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Sintering of particles of different sizes
Acta Materialia 55 (2007) 813–824 www.actamat-journals.com
Sintering of particles of different sizes H.N. Ch’ng a, Jingzhe Pan b
b,*
a School of Engineering, University of Surrey, Guildford GU2 7XH, UK Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
Received 21 March 2006; received in revised form 7 July 2006; accepted 7 July 2006 Available online 13 December 2006
Abstract A computer simulation study of the sintering process of cylindrical particles is presented following an earlier work by Pan et al. [Pan J, Le H, Kucherenko S, Yeomans JA. A model for the sintering of spherical particles of different sizes by solid state diffusion. Acta Metall 1998;46:4671–90]. The current paper focuses on the effect that the size of the particles has on the sintering kinetics. The computer simulation revealed a sophisticated behaviour for this apparently well-understood problem. Firstly, it is shown that a smaller particle can grow temporarily at the expense of its larger neighbours, in contradiction to conventional wisdom. There are two conditions for this to happen: (a) the particles must be nano-sized and (b) the smaller particle must be placed between two larger neighbours. Secondly, a pair of nanoparticles of different size can separate from each other in the later stage of their co-sintering process. Thirdly, the extent to which particles approach each other is strongly influenced by degree of constraint that the particles are subjected to by the neighbouring particles. This sophisticated behaviour revealed by the computer simulation questions the relevance of the oversimplistic sintering models to a real powder compact. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Modelling; Particles; Coarsening; Computer simulation
1. Introduction Sintering is a process in which fine particles in contact with each other form necks and bond together. The driving force for the neck formation is the reduction in the free surface energy of the particles. An elevated temperature is necessary for sintering so that the atoms become sufficiently mobile to migrate. The required sintering temperature is often much lower than the melting temperature of the powder material. Sintering is perhaps the only economical route to produce ceramic consumer products and advanced components on an industrial scale. For metallic, polymer and glass materials, sintering has produced many materials/components with unusual microstructures and properties, which cannot be obtained using other manufacturing technologies. One way to obtain such unusual properties *
Corresponding author. Tel.: +44 116 223 1093. E-mail address: [email protected] (J. Pan).
is to sinter nanostructured materials. Nano-sized powders of a wide range of materials are now reasonably economical to produce and abundantly available. The challenge is to maintain the nanostructure during sintering. This is because sintering also leads to particle coarsening – smaller particles coalesce with larger ones producing larger and larger grains. Many attempts have been made to control particle coarsening in nanopowder compacts [1]. For example, Chen and Wang [2] used a two-step heating schedule and Mayo used sinter-forging [1]. In general, it is still difficult to control particle coarsening and researchers often have to resort to impractical measures, such as applying a huge mechanical stress, in order to obtain a final product with nano-sized grains. On the other hand, it has been reported that fully dense nanograined material resists grain growth [3]. The strong tendency of nanoparticles to coarsen during sintering seems to be a consequence of particle agglomeration [1] rather than an intrinsic behaviour of the nanoparticles themselves. However, the intrinsic behaviour
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.015
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of nanoparticles during sintering is difficult to observe experimentally and is not well understood. The sintering of two particles of different sizes can be viewed from four different aspects: (i) the growth of the contact neck; (ii) the approach of the centres of the two particles; (iii) the redistribution of matter from the small particle to the large one and (iv) the migration of the interparticle boundary towards the smaller particle. The current understanding of the interplay between these four processes is poor. In early sintering models, particle coarsening was treated separately from neck growth and particle approach. On one hand, there are neck-growth and shrinkage laws, which were obtained on the assumption that all of the particles were the same size. On the other hand, there are coarsening laws that ignore neck-growth. The two are then mechanically linked together by updating the particle size in the shrinkage law using the particle coarsening law [4]. In fact, the four processes are strongly coupled to each other. Recent progress in numerical modelling techniques has made it possible to study the interplay using computer simulation. For example, Pan et al. [5,6] were probably the first to carry out computer simulations of the coupled shrinkage and particle coarsening, whilst Parhami et al. [7] developed an approximate model for the same process and Saitou [8] developed an analytical model for particle coarsening. However, in these models the interparticle boundary was assumed to remain straight, implying that the boundary had an infinitely large mobility. Zhang et al. [9] criticised this assumption and developed a robust mathematical model to study the interplay between neck-growth and particle coarsening. Wakai et al. [10] also carried out a series of computer simulations for the particle coarsening process, which allowed curved grain-boundaries. However, these models assumed that surface diffusion is the only operational mechanism for neck-growth. Such a system does not densify during sintering because the particles cannot approach each other. For submicron and smaller particles of a large range of materials, grain-boundary diffusion removes matter from the contact neck allowing the particles to approach each other [5]. Such a mechanism was not included in the models proposed by Zhang et al. [9] and Wakai et al. [10]. Recently Pan and co-workers [11–14] have developed a general framework to formulate sintering models which allows all possible matter redistribution mechanisms to be considered. A corresponding numerical scheme was also developed [15–17] to simulate the microstructural evolution and used to study some important issues in sintering [18]. In the current paper, this numerical tool is used to revisit the problem of co-sintering between two cylindrical particles of different sizes. For the first time, the full interaction between grain-boundary diffusion, surface diffusion and grain-boundary migration is studied in a sintering model. The current study has two main aims: (i) to understand how the absolute size of the particles affects the interplay between the three matter redistribution mechanisms; and
(ii) to explore whether a simple change in the particle arrangement affects significantly the way in which particles sinter. 2. Computer models for particle sintering Four different models are considered as shown in Fig. 1. In model I, a particle is in contact with another particle of twice its size. There is no interaction between this pair of particles and other particles. This model is more suited to the simulation of conditions in a particle synthesis process. In model II, the small particle is placed between two larger neighbours to simulate the condition in a particle compact. Symmetry is assumed across the centres of the two large particles. Models III and IV consist of identical particles and correspond to the particle arrangements in models I and II, respectively. There is no particle coarsening in these two models. They are used here as the basis of comparison with models I and II. The particles are in fact cylinders because the numerical models are two-dimensional in the current study. The driving force for the microstructural evolution is the reduction in the interfacial energy of the system, E, defined as Z Z E¼ cSS dA þ cSV dA; ð1Þ contact
surface
in which cSS and cSV represent the specific interfacial energies for the grain-boundary and the free surface, respectively. It is assumed that the particles are rigid and matter redistribution is achieved by solid-state diffusion along the free surface and the interparticle (grain) boundary and by the migration of the grain-boundary. At elevated temperatures, atoms can be removed from the grain-boundary by grain-boundary diffusion and deposited onto the particle surface by surface diffusion. Surface diffusion also redistributes matter between the large and small particles. At the same time atomic rearrangement normal to the grain-boundary leads to grain-boundary migration toward the small particle. These matter redistribution mechanisms can be represented by an ‘‘atomic migration * velocity’’ t in a thin layer of the particle surface and grain-boundary. A linear kinetic law *
*
t ¼MF
ð2Þ *
is assumed, in which F represents a thermodynamic driving force for the atomic migration, and M represents the mobility of the atoms which depends on the temperature and local environment in the thin layer. It can be shown that among all the possible velocity fields that satisfy matter conservation, the true one minimises the following functional [14,19]: Z 1 1 * * dE t t dV þ ; ð3Þ P¼ 2 V MX dt in which X is the atomic volume and the integration is over the entire volume of the solid particles.
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r0.5 +
815
c r
d c +
r0.5 +
+
r d
Model I
Model II
r + c d
r
+
+
c + d Model III
Model IV
Fig. 1. The four different models used in this study. Model I, two particles of different size; model II, a periodic row of alternating large and small particles; model III, two particles of the same size; model IV, a periodic row of particles of a same size. *
The velocity t in the thin layer of interface can be * decomposed into t k , which is parallel to the interface * and corresponds to solid state diffusion, and t ? , which is normal to the interface and corresponds to interface * migration. t ? does not exist on the particle surface because vapour evaporation and condensation are not considered here. The grain-boundary and surface diffusion * are often described using a diffusional flux j , which is defined as the volume of matter passing through a unit slab of the layer per unit time. Using d* to represent the * thickness of the diffusional layer, then j ¼ t k d. Noticing that dV = ddA in Eq. (3), the functional P can be rewritten as Z 1 * * P¼ j j dA interface 2M diff Z 1 * * dE ; ð4Þ t ? t ? dA þ þ 2M dt m grain-boundary in which two effective kinetic mobilities have been introduced M diff ¼ dMX
ð5Þ
and M m ¼ MX=d:
ð6Þ
The diffusion mobility, Mdiff, has very different values in the grain-boundary and at the particle surface due to the different atomic structures. The grain-boundary diffusion mobility is referred to as Mgb and the surface diffusion mobility is refereed to as Ms. In the numerical study, the ratio between the two mobilities is used as input data instead of the actual values in order for the numerical results to be more generally valid. The ratio of Ms/Mgb is varied over a wide range
to investigate the effect of the relative diffusion rate on the sintering kinetics. The size effect of the particle is controlled by a nondimensional group given by Mm ¼
M m r2 ; M gb
ð7Þ
in which r is the radius of the large particle. It is reasonable to assume that the atomic mobility M as defined in Eq. (2) * * is the same for t k and t ? . Eqs. (5) and (6) then lead to M m r 2 r 2 Mm ¼ ¼ : ð8Þ d M gb Eq. (8) is a significant expression because it dictates that M m depends only on the ratio between the particle size and the thickness of the diffusion layer. Svoboda and Riedel [20] defined a different parameter, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cSV M s xo ¼ ; ð9Þ cSS M m to study the same interplay between solid state diffusion and grain-boundary migration. For alumina they found that x0 = 0.07 lm by assuming cSV/cSS = 1. Here, using cSV/cSS = 3, which is a more suitable value [21], and Ms/Mgb = 50 at T = 1600 C (the temperature used by Svoboda and Riedel [20]) it is found that d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x0 cSS M gb =cSV M s ¼ 0:01 lm. This value of d is used in all of the following numerical cases. The estimated value of d is larger than expected for the thickness of the diffusion layer. This is because impurities at the grain-boundary affect Mgb and Mm differently and the value of d calculated from Eq. (8) using experimental data for Mgb and Mm
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should be considered as an empirical parameter rather than the actual thickness of the diffusion layer. From Eq. (8) and using d = 0.01 lm, it can be calculated that the relative mobility M m takes a value of 102 for nano-sized particles and a value of 104 for micron-sized particles. It will be shown in the later sections that this huge range of variation in the value of M m leads to a dramatic change in the sintering behaviour. The variational problem of dP = 0 is solved using the numerical scheme developed by Ch’ng and Pan [16,17].
The numerical scheme has been vigorously tested using a wide range of cases [16,17] and used to study the evolution of sophisticated microstructures [18]. Here one more test case is provided in order to demonstrate the validity of the variational model. Zhang and Schneibel [22] provided a finite difference solution to the problem of model III shown in Fig. 1. Here the same problem is solved using the variational model. Figs. 2 and 3 compare the numerical results to those obtained by Zhang and Schneibel [22]. Fig. 2 shows the neck size as a function of time, while Fig. 3 shows the distance between the centres of the two particles as a function of time. When presenting the numerical results, a non-dimensionalised time, t, has been used t ¼
cSV M gb t; r4
ð10Þ
in which r represents the initial radius of the particle. It can be seen from the two figures that the variational model agrees very well with the finite difference model by Zhang and Schneibel [22]. 3. Comparison of the four different models 3.1. Comparison of the two models of identical particles
Fig. 2. Neck size as a function of time obtained using the variational model (solid line) and by Zhang and Schneibel (discrete dots) [22]. cSV/cSS = 1.93 and Ms/Mgb = 1.
The very first observation from the computer simulation is that the two models of identical particles (models III and IV) produce very similar behaviour in terms of the neckgrowth and centre-to-centre approach, except that the equilibrium (final) values of the neck size and centreto-centre distance are different. Figs. 4 and 5 show the neck-size and the change in centre-to-centre distance as functions of the normalised time (as defined by Eq. (10)) obtained using the two different models for Ms/Mgb = 1. The evolution of the particle shape is also shown in the figures. A series of simulations were carried out using the two models and a large range of values of Ms/Mgb. The results show that increasing the ratio of Ms/Mgb, i.e. faster surface diffusion, accelerates neck growth but retards centreto-centre approach. This agrees with the long-established concept that fast surface diffusion retards densification. In all the cases, the two models predict very similar behaviour. These numerical results simply confirmed these existing understandings. 3.2. Comparison of the two models with particles of different size
Fig. 3. Centre-to-centre distance between the two particles obtained using the variational model (solid line) and by Zhang and Schneibel (dashed line) [22]. cSV/cSS = 1.93 and Ms/Mgb = 1.
The computer simulations using models I and II revealed several interesting phenomena. Figs. 6 and 7 show four different cases of the computer-simulated particle evolution. The difference between Figs. 6 and 7 is the value used for the normalised grain-boundary mobility M m (see Eq. (8)). For Fig. 6, M m was taken as 104, which corresponds to a radius of about 1 lm for the large particle. For Fig. 7, M m was taken as 100, which corresponds to a radius of about 100 nm for the large particle. All the other
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Fig. 4. Neck size and change in centre-to-centre distance as functions of time obtained using model III. c represents the neck size. d0 and d represent the initial and current distances between the two particle centres. cSV/cSS = 3 and Ms/Mgb = 1.
Fig. 5. Neck size and change in centre-to-centre distance as functions of time obtained using model IV. c represents the neck size. d0 and d represent the initial and current distances between the two particle centres. cSV/cSS = 3 and Ms/Mgb = 1.
input data in these models were taken to be the same as those in Section 3.1 (i.e., cSV/cSS = 3 and Ms/Mgb = 50). In order to show the effect of size with sintering time, a different normalised time (t*) from that of Eq. (10) is used when presenting these comparisons, such that: t ¼ cSS M gb t;
ð11Þ
which does not normalise the time with respect to the particle size. The comparison of Figs. 6(a.1)–(a.3) and 7(a.1)– (a.3) is interesting but not too surprising. In both cases, the
large particle consumed the small particle. For the micronsized particles, the diffusion distance is large so that the grain-boundary swept through the small particle before the diffusion process recovered the two particles into a single spherical one. For the nano-sized particles, the diffusion distance is short so that the two particles almost became circular before the grain-boundary swept through the small particle. The comparison of Figs. 6(b.1)–(b.3) and 7(b.1)–(b.6) is interesting. Fig. 6(b.1)–(b.3) shows an expected behaviour
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Fig. 6. Computer simulated shape evolution of micron-sized particles, rlarge = 1 lm. (a) Model II: (a.1) t* = 1.74 · 1029; (a.2) t* = 8.05 · 1029; (a.3) t* = 1.24 · 1028. (b) Model IV: (b.1) t* = 7.18 · 1031; (b.2) t* = 2.85 · 1029; (b.3) t* = 8.33 · 1029. Ms/Mgb = 50, and cSV/cSS = 3. The crosses show locations of the particle ‘‘centre’’.
whereby the large particles invaded the small one as soon as the sintering process started. The behaviour shown in Fig. 7(b.1)–(b.6), however, is more complicated. More intermediate profiles are provided in the figure to show the details of this case. For the nano-sized particles, the small particle developed a certain resistance to invasion. It quickly took up a concave shape and then drew material to itself from the two large particles by surface diffusion. This is possible because of the very short diffusion distance. At the very beginning of the sintering process, grainboundary diffusion quickly filled up the neck. While this fast filling occurred in both models II and IV, the symmetry condition in model IV made it possible for the small particle to take a concave shape. The small particle then grew, rather than shrinking, at the expense of the two large particles. Despite the initial resistance to coarsening, the small particle eventually disappeared as the two boundaries migrated towards each other. To demonstrate that surface diffusion plays a key role in this resistance to coarsening, the simulation of Fig. 7(b) was repeated using higher ratios of Ms/Mgb. Fig. 8 shows the normalised area of the large particle as a function of time obtained using two different values of Ms/Mgb. It can be seen from the figure that faster surface diffusion accelerates the transfer of material from the large particle to the small one and therefore retards par-
ticle coarsening. This is a significant conclusion. As discussed in Section 3.1, the established wisdom in ceramic processing is that surface diffusion retards densification and therefore should be avoided if one wants to achieve high density. As surface diffusion tends to dominate at lower sintering temperatures, this has led to the idea of using as fast a heating rate as possible (i.e. as fast as is compatible with the thermal shock resistance of the powder compact) in ceramic processing to achieve high density. Many attempts have been made to use fast heating rates to sinter nanopowder compacts but the results so far have been all negative [2]. The computer simulation result shown in Fig. 8 offers a possible explanation for these experimental findings. Fig. 9 presents the normalised area of the large particle as a function of time for a range of different sizes of the large particle. The log-scale is used for the normalised time so that results for different particle sizes can be presented on the same figure. The solid lines were obtained using model II and the dashed lines were obtained using model I. It can be seen from the figure that the two models predict a similar behaviour for micron-sized particles but very different behaviours for nano-sized particles. The changeover occurs between rlarge = 100 nm and rlarge = 1 lm. While in model I the large particle always grows, in model II if the
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Fig. 7. Computer simulated shape evolutions of nano-sized particles, rlarge = 100 nm. (a) Model II: (a.1) t* = 9.98 · 1034; (a.2) t* = 6.48 · 1032; (a.3) t* = 1.98 · 1031. (b) Model IV: (b.1) t* = 5.09 · 1033; (b.2) t* = 8.98 · 1032; (b.3) t* = 1.55 · 1031; (b.4) t* = 6.38 · 1031; (b.5) t* = 9.00 · 1031; (b.6) t* = 1.12 · 1030.
radius of the large particle is smaller than 100 nm, the large particle shrinks first and then grows. The continuum models used in the current paper are not valid for particles of a few nanometres in diameter; currently, molecular dynamic models are being used to study the phenomenon for the extremely small particles. The results for rlarge = 1 nm in the figure are therefore indicative only and provide a base for comparison for the molecular dynamic models. As each simulation took about 3 months to run on a PC, the simulation for the nano-sized particle was terminated as soon as the small particle started to shrink. The particle coarsening is often described by an Ostwald ripening theory [23] which predicts that dn increases linearly with time. Here, d is the average diameter of the particles and n depends on the matter transportation mechanism. The Ostwald ripening the-
ory is obviously unable to capture the complicated behaviour observed in this computer simulation. A detailed study of the numerical results of models I and II revealed another interesting phenomenon. Fig. 10 presents the change in the centre-to-centre distance, 1 d/d0, between the two particles as a function of time for a range of values of the large particle size. Here d and d0 represent the distance between the two particle centres at the current and the initial times, respectively, and the ‘‘centre’’ of a particle is taken as the current position of the initial centre of the circular particle. This centre moves rigidly with the entire particle and does not coincide with the mass centre of the particle except at t = 0. As shown in Figs. 6a and 7a, the as-defined centre of the small particle can find itself entirely located outside the particle because of the
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Fig. 8. Normalised area of the large particle as a function of time in model II for nano-sized particles (rlarge = 10 nm) using two different values of Ms/Mgb.
Fig. 9. Normalised area of the large particle as a function of time for a different sizes of the larger particle. The solid lines were obtained using model II and the dashed lines were obtained using model I. cSV/cSS = 3 and Ms/Mgb = 50. r represents the initial size of the large particle. (b.1)–(b.6) correspond to the stages of the particle evolution shown in Fig. 7(b.1)–(b.6).
grain-boundary migration (in the case of Fig. 6a) or matter redistribution from the small particle to the large one by surface diffusion (in the case of Fig. 7a). The advantage of defining the particle centre in this way is that the relative motion between the centres of the two particles in contact reflects grain-boundary diffusion. The reduction in the centre-to-centre distance leads to densification in a powder compact. From Fig. 10, it can be seen that in model II the two particles approach each other as expected. However, in model I the two particles approach each other first
and then separate from each other, indicating a change in direction of the grain-boundary diffusion. This behaviour contradicts the well-known scaling law due to Herring [24] for sintering controlled by grain-boundary diffusion, in which atoms always diffuse away from the particle boundary. Fig. 11 shows the particle geometries at the time of the change in the direction of diffusion for the three cases indicated by A, B and C in Fig. 10. The physical explanation behind the direction change is in fact very simple. When two particles are placed together, grain-boundary
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Fig. 10. Change in the centre-to-centre distance between the two particles as a function of time. d and d0 represent the distance between the two particle centres at the current and the initial times, respectively. The solid lines were obtained using model II and the dashed lines were obtained using model I. cSV/ cSS = 3 and Ms/Mgb = 50. r represents the initial radius of the large particle.
Fig. 11. Shapes of the two particles in model I at the moments of switching from particle approach to particle separation as indicated by A, B and C on the dashed lines in Fig. 10.
diffusion normally removes atoms from the boundary to fill up the contact neck because in doing so two free surfaces are replaced by a single grain-boundary so that the total free energy is reduced. This is clearly shown in Figs. 4 and 5 and is consistence with Herring’s model [24]. However, the particle shapes shown in Fig. 11 are very different from those in Figs. 4 and 5, and hence from the geometry considered by Herring. If one imagines a small liquid droplet that is dropped onto a flat solid surface, such that full wetting does not occur (i.e. the equilibrium contact angle is larger than 90), then the impact would force the liquid droplet into a shallow lens shape on the solid surface just like the shape of the smaller particle shown in Fig. 12a. The droplet would then quickly recover its equilibrium shape. The same driving force lies behind particle separa-
tion. Fig. 12 presents the contact angle as a function of time for the case of M m ¼ 102 in Fig. 10. In this case, cSV/cSS = 3, which corresponds to an equilibrium contact angle of 99.59. The figure shows two contact angles, one ‘‘local’’ and the other one ‘‘global’’. The local contact angle is a degree of freedom in the finite element formulation [17], which theoretically represents the contact angle if the triple point is enlarged by many times. The global contact angle represents the observed contact angle, which was obtained manually by drawing two tangent lines at the triple point and measuring the angle between them as shown in Fig. 12. It can be seen from Fig. 12 that while the local contact angle approaches the equilibrium one, the global contact angle reduces with time as a consequence of the fast redistribution of matter from the small particle to the large
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Fig. 12. Contact angle as a function of time in model I for rlarge = 1 nm (corresponding to case A in Fig. 11). For cSV/cSS = 3, the equilibrium contact angle is 99.59. The ‘‘local contact angle’’ is a degree of freedom obtained from the finite element analysis. Theoretically, this is the contact angle if the triple point is enlarged by many times. The ‘‘global contact angle’’ is the observed contact angle obtained manually by drawing two tangent lines at the triple point as shown in the figure. The divergence between the local and global contact angles shows the system evolves in a non-equilibrium manner.
one. The divergence between the two contact angles shows that the system evolves into a non-equilibrium. At a certain point, the small particle attempts a recovery, as the liquid droplet did after being dropped onto a flat surface, driven by the difference between the local and global contact angles. In the solid state, the recovery is achieved by a combination of inserting atoms onto the grain-boundary and free surface diffusion. The smaller the particle size, the more dominant the surface diffusion becomes. Therefore, only the nano-sized particles show this separation behaviour.
3.3. Comparison of the four models The changes in centre-to-centre distance predicted by the four models are compared in Figs. 13 and 14. An immediate observation is that the four models predict very different approach behaviour and that there is no clear pattern. Fig. 13 shows that for micron-sized particles, model I predicts the highest level of approach while model III predicts the least. Fig. 14 shows that the nanosized particles behave differently – model II predicts the closest particle approach and model I predicts the least.
Fig. 13. Change in centre-to-centre distance between the two particles as a function of time as predicted by models I–IV. M m ¼ 10; 000 which corresponds to rlarge = 1 lm. cSV/cSS = 3 and Ms/Mgb = 50.
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Fig. 14. Change in centre-to-centre distance between the two particles as a function of time as predicted by models I–IV. M m ¼ 1 which corresponds to rlarge = 10 nm. cSV/cSS = 3 and Ms/Mgb = 50.
In fact, as discussed previously, significant particle separation occurs in model I. For the nano-sized particles, constraint from neighbouring particles leads to a greater degree of approach as seen from Fig. 14. The same is not true for micron-sized particles as shown in Fig. 13. It is important to note that even at the very early stage of the sintering process, the four models predict very different behaviours. This shows the sensitivity of the particle approach to the details of the particle arrangement. Consequently the relevance of oversimplistic sintering models based on two identical particles with strong assumptions about contact geometry to the real powder compact should be questioned. 4. Conclusions The computer simulations presented here revealed several interesting phenomena that have not been reported before as far as the authors are aware. It was observed that a small particle can grow at the expense of its larger neighbours. There are two conditions for this to occur: (i) the particles have to be small enough for surface diffusion rather than grain-boundary migration to dominate the process, which typically requires the particles to be nano-sized and (ii) the small particle must be placed between two larger neighbours. This resistance to coarsening by the small particle is, however, only temporary and the small particle eventually disappears. The computer simulation showed that a faster surface diffusion can enhance the temporary resistance of the small particle to coarsening. In ceramic processing, it has been widely believed that fast surface diffusion retards densification and should be avoided. Surface diffusion tends to dominate at lower sintering temperatures. This has led to the idea of using very fast heating
rates in order to densify nano-sized powder compacts. However, these attempts have not been successful. The computer simulation offers a possible explanation for the failure of the technique. Another interesting observation in the computer simulation is that a pair of nanoparticles of different sizes can separate from each other in the later stage of their co-sintering process. The computer simulations also show that the sintering kinetics between two particles are very sensitive to the different ways in which the particles are connected to each other in the model. The current understanding of the interplay between the various sintering mechanisms is largely based on various oversimplistic models; the sophisticated behaviour presented here questions the relevance of the simplistic models to a real powder compact. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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[14] Pan J. Phil Mag Lett 2004;84(5):303–10. [15] Pan J, Cocks ACF, Kucherenko S. Proc Royal Soc London A 1997;453:2161–84. [16] Ch’ng HN, Pan J. J Comput Phys 2004;196:742–50. [17] Ch’ng HN, Pan J. J Comput Phys 2005;204(2):430–61. [18] Pan J, Ch’ng HN, Cocks ACF. Mech Mater 2005;37:705–21. [19] Cocks ACF, Gill SPA, Pan J. Adv Appl Mech 1999;36: 81–162.
[20] Svoboda J, Riedel H. Acta Metall 1992;40:2829–40. [21] Cottrell A. An Introduction to Metallurgy. London: Institute of Materials; 1995. p. 340–5. [22] Zhang W, Schneibel JH. Acta Metall 1995;43:4377. [23] Finsy R. On the critical radius in Ostwald ripening. Langmuir 2004;20:2975–6. [24] Herring C. In: Kingston WE, editor. The physics of powder metallurgy. New York, NY: McGraw-Hill; 1951. p. 143–79.
Sintering of particles of different sizes H.N. Ch’ng a, Jingzhe Pan b
b,*
a School of Engineering, University of Surrey, Guildford GU2 7XH, UK Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
Received 21 March 2006; received in revised form 7 July 2006; accepted 7 July 2006 Available online 13 December 2006
Abstract A computer simulation study of the sintering process of cylindrical particles is presented following an earlier work by Pan et al. [Pan J, Le H, Kucherenko S, Yeomans JA. A model for the sintering of spherical particles of different sizes by solid state diffusion. Acta Metall 1998;46:4671–90]. The current paper focuses on the effect that the size of the particles has on the sintering kinetics. The computer simulation revealed a sophisticated behaviour for this apparently well-understood problem. Firstly, it is shown that a smaller particle can grow temporarily at the expense of its larger neighbours, in contradiction to conventional wisdom. There are two conditions for this to happen: (a) the particles must be nano-sized and (b) the smaller particle must be placed between two larger neighbours. Secondly, a pair of nanoparticles of different size can separate from each other in the later stage of their co-sintering process. Thirdly, the extent to which particles approach each other is strongly influenced by degree of constraint that the particles are subjected to by the neighbouring particles. This sophisticated behaviour revealed by the computer simulation questions the relevance of the oversimplistic sintering models to a real powder compact. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Modelling; Particles; Coarsening; Computer simulation
1. Introduction Sintering is a process in which fine particles in contact with each other form necks and bond together. The driving force for the neck formation is the reduction in the free surface energy of the particles. An elevated temperature is necessary for sintering so that the atoms become sufficiently mobile to migrate. The required sintering temperature is often much lower than the melting temperature of the powder material. Sintering is perhaps the only economical route to produce ceramic consumer products and advanced components on an industrial scale. For metallic, polymer and glass materials, sintering has produced many materials/components with unusual microstructures and properties, which cannot be obtained using other manufacturing technologies. One way to obtain such unusual properties *
Corresponding author. Tel.: +44 116 223 1093. E-mail address: [email protected] (J. Pan).
is to sinter nanostructured materials. Nano-sized powders of a wide range of materials are now reasonably economical to produce and abundantly available. The challenge is to maintain the nanostructure during sintering. This is because sintering also leads to particle coarsening – smaller particles coalesce with larger ones producing larger and larger grains. Many attempts have been made to control particle coarsening in nanopowder compacts [1]. For example, Chen and Wang [2] used a two-step heating schedule and Mayo used sinter-forging [1]. In general, it is still difficult to control particle coarsening and researchers often have to resort to impractical measures, such as applying a huge mechanical stress, in order to obtain a final product with nano-sized grains. On the other hand, it has been reported that fully dense nanograined material resists grain growth [3]. The strong tendency of nanoparticles to coarsen during sintering seems to be a consequence of particle agglomeration [1] rather than an intrinsic behaviour of the nanoparticles themselves. However, the intrinsic behaviour
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.015
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of nanoparticles during sintering is difficult to observe experimentally and is not well understood. The sintering of two particles of different sizes can be viewed from four different aspects: (i) the growth of the contact neck; (ii) the approach of the centres of the two particles; (iii) the redistribution of matter from the small particle to the large one and (iv) the migration of the interparticle boundary towards the smaller particle. The current understanding of the interplay between these four processes is poor. In early sintering models, particle coarsening was treated separately from neck growth and particle approach. On one hand, there are neck-growth and shrinkage laws, which were obtained on the assumption that all of the particles were the same size. On the other hand, there are coarsening laws that ignore neck-growth. The two are then mechanically linked together by updating the particle size in the shrinkage law using the particle coarsening law [4]. In fact, the four processes are strongly coupled to each other. Recent progress in numerical modelling techniques has made it possible to study the interplay using computer simulation. For example, Pan et al. [5,6] were probably the first to carry out computer simulations of the coupled shrinkage and particle coarsening, whilst Parhami et al. [7] developed an approximate model for the same process and Saitou [8] developed an analytical model for particle coarsening. However, in these models the interparticle boundary was assumed to remain straight, implying that the boundary had an infinitely large mobility. Zhang et al. [9] criticised this assumption and developed a robust mathematical model to study the interplay between neck-growth and particle coarsening. Wakai et al. [10] also carried out a series of computer simulations for the particle coarsening process, which allowed curved grain-boundaries. However, these models assumed that surface diffusion is the only operational mechanism for neck-growth. Such a system does not densify during sintering because the particles cannot approach each other. For submicron and smaller particles of a large range of materials, grain-boundary diffusion removes matter from the contact neck allowing the particles to approach each other [5]. Such a mechanism was not included in the models proposed by Zhang et al. [9] and Wakai et al. [10]. Recently Pan and co-workers [11–14] have developed a general framework to formulate sintering models which allows all possible matter redistribution mechanisms to be considered. A corresponding numerical scheme was also developed [15–17] to simulate the microstructural evolution and used to study some important issues in sintering [18]. In the current paper, this numerical tool is used to revisit the problem of co-sintering between two cylindrical particles of different sizes. For the first time, the full interaction between grain-boundary diffusion, surface diffusion and grain-boundary migration is studied in a sintering model. The current study has two main aims: (i) to understand how the absolute size of the particles affects the interplay between the three matter redistribution mechanisms; and
(ii) to explore whether a simple change in the particle arrangement affects significantly the way in which particles sinter. 2. Computer models for particle sintering Four different models are considered as shown in Fig. 1. In model I, a particle is in contact with another particle of twice its size. There is no interaction between this pair of particles and other particles. This model is more suited to the simulation of conditions in a particle synthesis process. In model II, the small particle is placed between two larger neighbours to simulate the condition in a particle compact. Symmetry is assumed across the centres of the two large particles. Models III and IV consist of identical particles and correspond to the particle arrangements in models I and II, respectively. There is no particle coarsening in these two models. They are used here as the basis of comparison with models I and II. The particles are in fact cylinders because the numerical models are two-dimensional in the current study. The driving force for the microstructural evolution is the reduction in the interfacial energy of the system, E, defined as Z Z E¼ cSS dA þ cSV dA; ð1Þ contact
surface
in which cSS and cSV represent the specific interfacial energies for the grain-boundary and the free surface, respectively. It is assumed that the particles are rigid and matter redistribution is achieved by solid-state diffusion along the free surface and the interparticle (grain) boundary and by the migration of the grain-boundary. At elevated temperatures, atoms can be removed from the grain-boundary by grain-boundary diffusion and deposited onto the particle surface by surface diffusion. Surface diffusion also redistributes matter between the large and small particles. At the same time atomic rearrangement normal to the grain-boundary leads to grain-boundary migration toward the small particle. These matter redistribution mechanisms can be represented by an ‘‘atomic migration * velocity’’ t in a thin layer of the particle surface and grain-boundary. A linear kinetic law *
*
t ¼MF
ð2Þ *
is assumed, in which F represents a thermodynamic driving force for the atomic migration, and M represents the mobility of the atoms which depends on the temperature and local environment in the thin layer. It can be shown that among all the possible velocity fields that satisfy matter conservation, the true one minimises the following functional [14,19]: Z 1 1 * * dE t t dV þ ; ð3Þ P¼ 2 V MX dt in which X is the atomic volume and the integration is over the entire volume of the solid particles.
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r0.5 +
815
c r
d c +
r0.5 +
+
r d
Model I
Model II
r + c d
r
+
+
c + d Model III
Model IV
Fig. 1. The four different models used in this study. Model I, two particles of different size; model II, a periodic row of alternating large and small particles; model III, two particles of the same size; model IV, a periodic row of particles of a same size. *
The velocity t in the thin layer of interface can be * decomposed into t k , which is parallel to the interface * and corresponds to solid state diffusion, and t ? , which is normal to the interface and corresponds to interface * migration. t ? does not exist on the particle surface because vapour evaporation and condensation are not considered here. The grain-boundary and surface diffusion * are often described using a diffusional flux j , which is defined as the volume of matter passing through a unit slab of the layer per unit time. Using d* to represent the * thickness of the diffusional layer, then j ¼ t k d. Noticing that dV = ddA in Eq. (3), the functional P can be rewritten as Z 1 * * P¼ j j dA interface 2M diff Z 1 * * dE ; ð4Þ t ? t ? dA þ þ 2M dt m grain-boundary in which two effective kinetic mobilities have been introduced M diff ¼ dMX
ð5Þ
and M m ¼ MX=d:
ð6Þ
The diffusion mobility, Mdiff, has very different values in the grain-boundary and at the particle surface due to the different atomic structures. The grain-boundary diffusion mobility is referred to as Mgb and the surface diffusion mobility is refereed to as Ms. In the numerical study, the ratio between the two mobilities is used as input data instead of the actual values in order for the numerical results to be more generally valid. The ratio of Ms/Mgb is varied over a wide range
to investigate the effect of the relative diffusion rate on the sintering kinetics. The size effect of the particle is controlled by a nondimensional group given by Mm ¼
M m r2 ; M gb
ð7Þ
in which r is the radius of the large particle. It is reasonable to assume that the atomic mobility M as defined in Eq. (2) * * is the same for t k and t ? . Eqs. (5) and (6) then lead to M m r 2 r 2 Mm ¼ ¼ : ð8Þ d M gb Eq. (8) is a significant expression because it dictates that M m depends only on the ratio between the particle size and the thickness of the diffusion layer. Svoboda and Riedel [20] defined a different parameter, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cSV M s xo ¼ ; ð9Þ cSS M m to study the same interplay between solid state diffusion and grain-boundary migration. For alumina they found that x0 = 0.07 lm by assuming cSV/cSS = 1. Here, using cSV/cSS = 3, which is a more suitable value [21], and Ms/Mgb = 50 at T = 1600 C (the temperature used by Svoboda and Riedel [20]) it is found that d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x0 cSS M gb =cSV M s ¼ 0:01 lm. This value of d is used in all of the following numerical cases. The estimated value of d is larger than expected for the thickness of the diffusion layer. This is because impurities at the grain-boundary affect Mgb and Mm differently and the value of d calculated from Eq. (8) using experimental data for Mgb and Mm
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should be considered as an empirical parameter rather than the actual thickness of the diffusion layer. From Eq. (8) and using d = 0.01 lm, it can be calculated that the relative mobility M m takes a value of 102 for nano-sized particles and a value of 104 for micron-sized particles. It will be shown in the later sections that this huge range of variation in the value of M m leads to a dramatic change in the sintering behaviour. The variational problem of dP = 0 is solved using the numerical scheme developed by Ch’ng and Pan [16,17].
The numerical scheme has been vigorously tested using a wide range of cases [16,17] and used to study the evolution of sophisticated microstructures [18]. Here one more test case is provided in order to demonstrate the validity of the variational model. Zhang and Schneibel [22] provided a finite difference solution to the problem of model III shown in Fig. 1. Here the same problem is solved using the variational model. Figs. 2 and 3 compare the numerical results to those obtained by Zhang and Schneibel [22]. Fig. 2 shows the neck size as a function of time, while Fig. 3 shows the distance between the centres of the two particles as a function of time. When presenting the numerical results, a non-dimensionalised time, t, has been used t ¼
cSV M gb t; r4
ð10Þ
in which r represents the initial radius of the particle. It can be seen from the two figures that the variational model agrees very well with the finite difference model by Zhang and Schneibel [22]. 3. Comparison of the four different models 3.1. Comparison of the two models of identical particles
Fig. 2. Neck size as a function of time obtained using the variational model (solid line) and by Zhang and Schneibel (discrete dots) [22]. cSV/cSS = 1.93 and Ms/Mgb = 1.
The very first observation from the computer simulation is that the two models of identical particles (models III and IV) produce very similar behaviour in terms of the neckgrowth and centre-to-centre approach, except that the equilibrium (final) values of the neck size and centreto-centre distance are different. Figs. 4 and 5 show the neck-size and the change in centre-to-centre distance as functions of the normalised time (as defined by Eq. (10)) obtained using the two different models for Ms/Mgb = 1. The evolution of the particle shape is also shown in the figures. A series of simulations were carried out using the two models and a large range of values of Ms/Mgb. The results show that increasing the ratio of Ms/Mgb, i.e. faster surface diffusion, accelerates neck growth but retards centreto-centre approach. This agrees with the long-established concept that fast surface diffusion retards densification. In all the cases, the two models predict very similar behaviour. These numerical results simply confirmed these existing understandings. 3.2. Comparison of the two models with particles of different size
Fig. 3. Centre-to-centre distance between the two particles obtained using the variational model (solid line) and by Zhang and Schneibel (dashed line) [22]. cSV/cSS = 1.93 and Ms/Mgb = 1.
The computer simulations using models I and II revealed several interesting phenomena. Figs. 6 and 7 show four different cases of the computer-simulated particle evolution. The difference between Figs. 6 and 7 is the value used for the normalised grain-boundary mobility M m (see Eq. (8)). For Fig. 6, M m was taken as 104, which corresponds to a radius of about 1 lm for the large particle. For Fig. 7, M m was taken as 100, which corresponds to a radius of about 100 nm for the large particle. All the other
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Fig. 4. Neck size and change in centre-to-centre distance as functions of time obtained using model III. c represents the neck size. d0 and d represent the initial and current distances between the two particle centres. cSV/cSS = 3 and Ms/Mgb = 1.
Fig. 5. Neck size and change in centre-to-centre distance as functions of time obtained using model IV. c represents the neck size. d0 and d represent the initial and current distances between the two particle centres. cSV/cSS = 3 and Ms/Mgb = 1.
input data in these models were taken to be the same as those in Section 3.1 (i.e., cSV/cSS = 3 and Ms/Mgb = 50). In order to show the effect of size with sintering time, a different normalised time (t*) from that of Eq. (10) is used when presenting these comparisons, such that: t ¼ cSS M gb t;
ð11Þ
which does not normalise the time with respect to the particle size. The comparison of Figs. 6(a.1)–(a.3) and 7(a.1)– (a.3) is interesting but not too surprising. In both cases, the
large particle consumed the small particle. For the micronsized particles, the diffusion distance is large so that the grain-boundary swept through the small particle before the diffusion process recovered the two particles into a single spherical one. For the nano-sized particles, the diffusion distance is short so that the two particles almost became circular before the grain-boundary swept through the small particle. The comparison of Figs. 6(b.1)–(b.3) and 7(b.1)–(b.6) is interesting. Fig. 6(b.1)–(b.3) shows an expected behaviour
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Fig. 6. Computer simulated shape evolution of micron-sized particles, rlarge = 1 lm. (a) Model II: (a.1) t* = 1.74 · 1029; (a.2) t* = 8.05 · 1029; (a.3) t* = 1.24 · 1028. (b) Model IV: (b.1) t* = 7.18 · 1031; (b.2) t* = 2.85 · 1029; (b.3) t* = 8.33 · 1029. Ms/Mgb = 50, and cSV/cSS = 3. The crosses show locations of the particle ‘‘centre’’.
whereby the large particles invaded the small one as soon as the sintering process started. The behaviour shown in Fig. 7(b.1)–(b.6), however, is more complicated. More intermediate profiles are provided in the figure to show the details of this case. For the nano-sized particles, the small particle developed a certain resistance to invasion. It quickly took up a concave shape and then drew material to itself from the two large particles by surface diffusion. This is possible because of the very short diffusion distance. At the very beginning of the sintering process, grainboundary diffusion quickly filled up the neck. While this fast filling occurred in both models II and IV, the symmetry condition in model IV made it possible for the small particle to take a concave shape. The small particle then grew, rather than shrinking, at the expense of the two large particles. Despite the initial resistance to coarsening, the small particle eventually disappeared as the two boundaries migrated towards each other. To demonstrate that surface diffusion plays a key role in this resistance to coarsening, the simulation of Fig. 7(b) was repeated using higher ratios of Ms/Mgb. Fig. 8 shows the normalised area of the large particle as a function of time obtained using two different values of Ms/Mgb. It can be seen from the figure that faster surface diffusion accelerates the transfer of material from the large particle to the small one and therefore retards par-
ticle coarsening. This is a significant conclusion. As discussed in Section 3.1, the established wisdom in ceramic processing is that surface diffusion retards densification and therefore should be avoided if one wants to achieve high density. As surface diffusion tends to dominate at lower sintering temperatures, this has led to the idea of using as fast a heating rate as possible (i.e. as fast as is compatible with the thermal shock resistance of the powder compact) in ceramic processing to achieve high density. Many attempts have been made to use fast heating rates to sinter nanopowder compacts but the results so far have been all negative [2]. The computer simulation result shown in Fig. 8 offers a possible explanation for these experimental findings. Fig. 9 presents the normalised area of the large particle as a function of time for a range of different sizes of the large particle. The log-scale is used for the normalised time so that results for different particle sizes can be presented on the same figure. The solid lines were obtained using model II and the dashed lines were obtained using model I. It can be seen from the figure that the two models predict a similar behaviour for micron-sized particles but very different behaviours for nano-sized particles. The changeover occurs between rlarge = 100 nm and rlarge = 1 lm. While in model I the large particle always grows, in model II if the
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Fig. 7. Computer simulated shape evolutions of nano-sized particles, rlarge = 100 nm. (a) Model II: (a.1) t* = 9.98 · 1034; (a.2) t* = 6.48 · 1032; (a.3) t* = 1.98 · 1031. (b) Model IV: (b.1) t* = 5.09 · 1033; (b.2) t* = 8.98 · 1032; (b.3) t* = 1.55 · 1031; (b.4) t* = 6.38 · 1031; (b.5) t* = 9.00 · 1031; (b.6) t* = 1.12 · 1030.
radius of the large particle is smaller than 100 nm, the large particle shrinks first and then grows. The continuum models used in the current paper are not valid for particles of a few nanometres in diameter; currently, molecular dynamic models are being used to study the phenomenon for the extremely small particles. The results for rlarge = 1 nm in the figure are therefore indicative only and provide a base for comparison for the molecular dynamic models. As each simulation took about 3 months to run on a PC, the simulation for the nano-sized particle was terminated as soon as the small particle started to shrink. The particle coarsening is often described by an Ostwald ripening theory [23] which predicts that dn increases linearly with time. Here, d is the average diameter of the particles and n depends on the matter transportation mechanism. The Ostwald ripening the-
ory is obviously unable to capture the complicated behaviour observed in this computer simulation. A detailed study of the numerical results of models I and II revealed another interesting phenomenon. Fig. 10 presents the change in the centre-to-centre distance, 1 d/d0, between the two particles as a function of time for a range of values of the large particle size. Here d and d0 represent the distance between the two particle centres at the current and the initial times, respectively, and the ‘‘centre’’ of a particle is taken as the current position of the initial centre of the circular particle. This centre moves rigidly with the entire particle and does not coincide with the mass centre of the particle except at t = 0. As shown in Figs. 6a and 7a, the as-defined centre of the small particle can find itself entirely located outside the particle because of the
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Fig. 8. Normalised area of the large particle as a function of time in model II for nano-sized particles (rlarge = 10 nm) using two different values of Ms/Mgb.
Fig. 9. Normalised area of the large particle as a function of time for a different sizes of the larger particle. The solid lines were obtained using model II and the dashed lines were obtained using model I. cSV/cSS = 3 and Ms/Mgb = 50. r represents the initial size of the large particle. (b.1)–(b.6) correspond to the stages of the particle evolution shown in Fig. 7(b.1)–(b.6).
grain-boundary migration (in the case of Fig. 6a) or matter redistribution from the small particle to the large one by surface diffusion (in the case of Fig. 7a). The advantage of defining the particle centre in this way is that the relative motion between the centres of the two particles in contact reflects grain-boundary diffusion. The reduction in the centre-to-centre distance leads to densification in a powder compact. From Fig. 10, it can be seen that in model II the two particles approach each other as expected. However, in model I the two particles approach each other first
and then separate from each other, indicating a change in direction of the grain-boundary diffusion. This behaviour contradicts the well-known scaling law due to Herring [24] for sintering controlled by grain-boundary diffusion, in which atoms always diffuse away from the particle boundary. Fig. 11 shows the particle geometries at the time of the change in the direction of diffusion for the three cases indicated by A, B and C in Fig. 10. The physical explanation behind the direction change is in fact very simple. When two particles are placed together, grain-boundary
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Fig. 10. Change in the centre-to-centre distance between the two particles as a function of time. d and d0 represent the distance between the two particle centres at the current and the initial times, respectively. The solid lines were obtained using model II and the dashed lines were obtained using model I. cSV/ cSS = 3 and Ms/Mgb = 50. r represents the initial radius of the large particle.
Fig. 11. Shapes of the two particles in model I at the moments of switching from particle approach to particle separation as indicated by A, B and C on the dashed lines in Fig. 10.
diffusion normally removes atoms from the boundary to fill up the contact neck because in doing so two free surfaces are replaced by a single grain-boundary so that the total free energy is reduced. This is clearly shown in Figs. 4 and 5 and is consistence with Herring’s model [24]. However, the particle shapes shown in Fig. 11 are very different from those in Figs. 4 and 5, and hence from the geometry considered by Herring. If one imagines a small liquid droplet that is dropped onto a flat solid surface, such that full wetting does not occur (i.e. the equilibrium contact angle is larger than 90), then the impact would force the liquid droplet into a shallow lens shape on the solid surface just like the shape of the smaller particle shown in Fig. 12a. The droplet would then quickly recover its equilibrium shape. The same driving force lies behind particle separa-
tion. Fig. 12 presents the contact angle as a function of time for the case of M m ¼ 102 in Fig. 10. In this case, cSV/cSS = 3, which corresponds to an equilibrium contact angle of 99.59. The figure shows two contact angles, one ‘‘local’’ and the other one ‘‘global’’. The local contact angle is a degree of freedom in the finite element formulation [17], which theoretically represents the contact angle if the triple point is enlarged by many times. The global contact angle represents the observed contact angle, which was obtained manually by drawing two tangent lines at the triple point and measuring the angle between them as shown in Fig. 12. It can be seen from Fig. 12 that while the local contact angle approaches the equilibrium one, the global contact angle reduces with time as a consequence of the fast redistribution of matter from the small particle to the large
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Fig. 12. Contact angle as a function of time in model I for rlarge = 1 nm (corresponding to case A in Fig. 11). For cSV/cSS = 3, the equilibrium contact angle is 99.59. The ‘‘local contact angle’’ is a degree of freedom obtained from the finite element analysis. Theoretically, this is the contact angle if the triple point is enlarged by many times. The ‘‘global contact angle’’ is the observed contact angle obtained manually by drawing two tangent lines at the triple point as shown in the figure. The divergence between the local and global contact angles shows the system evolves in a non-equilibrium manner.
one. The divergence between the two contact angles shows that the system evolves into a non-equilibrium. At a certain point, the small particle attempts a recovery, as the liquid droplet did after being dropped onto a flat surface, driven by the difference between the local and global contact angles. In the solid state, the recovery is achieved by a combination of inserting atoms onto the grain-boundary and free surface diffusion. The smaller the particle size, the more dominant the surface diffusion becomes. Therefore, only the nano-sized particles show this separation behaviour.
3.3. Comparison of the four models The changes in centre-to-centre distance predicted by the four models are compared in Figs. 13 and 14. An immediate observation is that the four models predict very different approach behaviour and that there is no clear pattern. Fig. 13 shows that for micron-sized particles, model I predicts the highest level of approach while model III predicts the least. Fig. 14 shows that the nanosized particles behave differently – model II predicts the closest particle approach and model I predicts the least.
Fig. 13. Change in centre-to-centre distance between the two particles as a function of time as predicted by models I–IV. M m ¼ 10; 000 which corresponds to rlarge = 1 lm. cSV/cSS = 3 and Ms/Mgb = 50.
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Fig. 14. Change in centre-to-centre distance between the two particles as a function of time as predicted by models I–IV. M m ¼ 1 which corresponds to rlarge = 10 nm. cSV/cSS = 3 and Ms/Mgb = 50.
In fact, as discussed previously, significant particle separation occurs in model I. For the nano-sized particles, constraint from neighbouring particles leads to a greater degree of approach as seen from Fig. 14. The same is not true for micron-sized particles as shown in Fig. 13. It is important to note that even at the very early stage of the sintering process, the four models predict very different behaviours. This shows the sensitivity of the particle approach to the details of the particle arrangement. Consequently the relevance of oversimplistic sintering models based on two identical particles with strong assumptions about contact geometry to the real powder compact should be questioned. 4. Conclusions The computer simulations presented here revealed several interesting phenomena that have not been reported before as far as the authors are aware. It was observed that a small particle can grow at the expense of its larger neighbours. There are two conditions for this to occur: (i) the particles have to be small enough for surface diffusion rather than grain-boundary migration to dominate the process, which typically requires the particles to be nano-sized and (ii) the small particle must be placed between two larger neighbours. This resistance to coarsening by the small particle is, however, only temporary and the small particle eventually disappears. The computer simulation showed that a faster surface diffusion can enhance the temporary resistance of the small particle to coarsening. In ceramic processing, it has been widely believed that fast surface diffusion retards densification and should be avoided. Surface diffusion tends to dominate at lower sintering temperatures. This has led to the idea of using very fast heating
rates in order to densify nano-sized powder compacts. However, these attempts have not been successful. The computer simulation offers a possible explanation for the failure of the technique. Another interesting observation in the computer simulation is that a pair of nanoparticles of different sizes can separate from each other in the later stage of their co-sintering process. The computer simulations also show that the sintering kinetics between two particles are very sensitive to the different ways in which the particles are connected to each other in the model. The current understanding of the interplay between the various sintering mechanisms is largely based on various oversimplistic models; the sophisticated behaviour presented here questions the relevance of the simplistic models to a real powder compact. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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