SCIENTIFIC RESEARCH AND DEVELOPMENT

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1 DOI /s Refractories and Industrial Ceramics Vol. 55, No. 5, January, 2015 SCIENTIFIC RESEARCH AND DEVELOPMENT POWDER COMPACT STRUCTURE. PART 3. THEORETICAL ANALYSIS OF SINTERING IN POWDER COMPACTS WITH INHOMOGENEOUS POROSITY 1 A. V. Galakhov 2,3 Translated from Novye Ogneupory, No. 9, pp , September Original article submitted July 17, Publications devoted to theoretical analysis of powder compact sintering are reviewed taking account of particle packing inhomogeneity within it. The methods used for resolving the problem are conditionally divided into analytical and numerical. It is shown that in order to obtain results agreeing with experimental observations more extensive possibilities may be realized using numerical methods in combination with computer modeling of particle packing. Examples are provided of implementing the method applied to oxide ceramic material sintering. Keywords: powder compact, structure, inhomogeneity, sintering, analytical methods, numerical methods. In sections of the present review published previously information about particle packing inhomogeneity in a powder compact, criteria for its evaluation, and reasons its generation have been collected and presented. Results have been provided in part 1 of the clearest experimental work for the effect of this factor on sintering [1]. In part 2 [2] the main attention was devoted to reviewing methods for improving powder compact structural inhomogeneity, which are conditionally separated into production and structural. Part 3 is devoted to a review of theoretical methods used for evaluating the effect of powder compact structural inhomogeneity on sintering and structure formation within sintered material. Theoretical description of sintering occupies a considerable place in contemporary materials science, or more precisely its sub-division powder metallurgy. Today there are two main fundamentally differing approaches to describing sintering; classical sintering theory [3] and phenomenological [4, 5]. The first rests on physical constants at the level of individual particle reaction. Phenomenological theory 1 Part 1 of the article published in Novye Ogneupory No. 5 of 2014, Part 2 in No. 6 of FGBUN A. A. Baikov Institute of Metallurgy and Materials Science, Russian Academy of Sciences, Moscow, Russia. 3 aleksander.galakhov@yandex.ru based on continuum solid mechanical equations (viscous medium) requires for its implementation existence of empirical coefficients used for describing macroscopic problems of powder technology. The author of this review has attempted to present as much as research results as possible for theoretical evaluation of the effect of powder compact structural inhomogeneity on sintering using both physical and phenomenological theory. In addition, in work on this theme one more tendency is considered, requiring conditional classification. Whereas the majority of early research for studying transformation of a pore space during sintering was performed by means of analytical (often cumbersome and complicated) relationships, subsequently numerical methods are used more often for this purpose. In spite of the fact that in some cases in order to obtain results computer technology is used actively, separation of material into two sub-sections, i.e., analytical and numerical methods, is correct. ANALYTICAL METHODS A quality criterion of powder compact structural inhomogeneity with a monomodal pore size distribution is the width of this distribution [1]. In analyzing sintering in such compacts it is convenient to use a functional form of this distribution. A similar approach has been demonstrated Springer Science+Business Media New York

2 Powder Compact Structure. Part in publications [6 8]. An example is given below of use of an analytical relationship describing transformation of the form monomodal pore size distribution during sintering [6]. Let the distribution of pore volume within a powder compact be described by a normalized distribution function within space of pore radii r and sintering time t: r F(, r t) f(, r t) dr. (1) 0 In physical sintering theory the mechanism of pore shrinkage is treated as a result of diffusion mass transfer in an assembly of touching particles. The moving force of this process is the concentration gradient of elementary mass carriers, often vacancies, within local powder compact regions. The process of diffusion healing of pores obeys a relationship dr dt J v, (2) where J v is vacancy flow density towards a pore of radius r; is vacancy volume. The density of vacancy flow depends on the reciprocal position of pores, and presence of different mass transfer paths towards a shrinking pore. A situation is considered in [6] when within the vicinity of a pore of radius r there are two forms of vacancy flow: grain boundary and approaching a neighboring pore. For these conditions Eq. (2), describing a change in pore radius in time, may be transformed to a form dr dt r d D v 1 1 kt rd r r s s where is current integral compact porosity; r s is average pore radius in a distribution; is particle material surface energy; D v is vacancy volumetric diffusion coefficient; d is average particle diameter; T is temperature; k is Boltzmann constant. The time change of natural pore distribution form may be obtained using a condition of distribution function continuity (1) with a size space [7]: df dt (3) v f f r r, (4) where v dr/dt; f f(r, t). Substitution of Eq. (3) in expression (4) gives an expression suitable for describing distribution shape change. On normal coordinates (relative average pore radius r so in an original distribution) it appears as follows [6]: f B 1 f r B 2 f 1 (5) t rd r B r 1 where B 1 d / r s 0; B ( 2 d )/ r A = 2D s0; v /(kt ); r r/ r s 0; d d / r s0; t ta / r 3. s Fig. 1. Change of pore volume distribution with respect to radius in time. In publication [6] by means of Eq. (5) the effect of pore size distribution width on sintering kinetics was analyzed. The width of the original pore size distribution was evaluated by the value of variation coefficient: v 0 = 0 /r s0 ( 0 is standard deviation for the original distribution). Normal distributions with coefficients of variation v 0 = 0.1, v 0 = 0.3, and v 0 = 0.5, and average radius r s0 = 1 are used for starting calculations. The value of compact original porosity 0 = 0.5, and average powder particle size d = 0.3. The time dependent transformation of different width distribution is shown in Fig. 1. Sintering kinetics were specified by the change in time of relative compact density. Calculated kinetic compaction curves (increase in density in time), corresponding to these distributions, are shown in Fig. 2, from which is seen a compact with a narrow pore size volume distribution densifiess more rapidly. However, in the initial sintering stage the rate of material compaction with an initially broad distribution is higher. This is explained by the fact that in the initial sintering stage within material with a broad distribution finer pores sinter more readily. In the concluding stage of sintering the rate of material compaction with an initially broad distribution falls sharply. Due to presence of a broad pore distribution, differing strongly with respect to size, in such a compact there is rapid diffusion pumping of vacancies from fine pores into coarse pores. As a result of this in the early sintering stage a coarse grained structure forms within it, whose sintering is difficult. This is seen from a change in maximum pore radius in distributions r max (Fig. 3). Whereas for material with a narrow distribution with an increase in density the size of a maximum pores decreases, in material with a broad distribution it increases. Pore growth within a sintered compact is not contradicted by experimental observation [1]. By using the results obtained it is possible to formulate general features of the effect of pore distribution width on sintering of powder moldings with an identical particle size. First, with sintering of a compact with a narrow pore size distribution densification of a compact commences sooner, proceeds more rapidly, and ceases sooner, than on sintering a compact with an initially broad distribution; in other words, the time interval of sintering for material exhibiting a distri-

3 450 A. V. Galakhov Fig. 2. Calculated compaction kinetic curves in distribution with a different original width. Fig. 5. Dependence of density on sintering time (a) and average grain size on achieved density (b ) for model compacts with a different pore size distribution width [9]. Fig. 3. Change in time of maximum pore radius r max in distributions with a different original width. Fig. 4. Kelvin tetradodecahedron and ratios for particle volume V, specific surface S, and grain size for it G; l r is rib length. bution with a low coefficient of variation, is narrow, but the densification rate within this range is high. Second, during compact sintering with a narrow size distribution pores decrease uniformly. For a compact with a broad distribution coarse pore growth is possible. These results may serve as a basis for some practical recommendations for sintering powder moldings. In particular, since a molding with a narrow pore size distribution exhibits a narrow sintering range, within which densification proceeds rapidly and at high rates, it is desirable to use them in sintering large objects. In this case a significant temperature gradient is unavoidable throughout a molding volume, and this may lead to object breakage due to difference shrinkage rate throughout the volume. For small objects, when the temperature gradient is not so large, use of compacts with a narrow pore size distribution makes it possible to accelerate sintering considerably and increase material final density. In work [6] considered above, analysis was carried out using functional pore size distribution, but without considering features of pore space geometry. In order to consider this factor there is often use of a simplified pore space model. Such an analytical model has been proposed for example in [9]. The geometric space of a porous compact is considered as a collection of Kelvin dodecahedra (Fig., 4); the convenience of this geometric model consists of a possibility of determining integral structures characteristics by simple analytical relationships (see Fig. 4). Calculated kinetic curves are shown in Fig. 5, specifying the effect of particle distribution width in an original packing on its compaction and increase in average grain size during sintering [9]. Compacts with a narrow pore size distribution (as a rule this is maximum density compacts of monodispersed particles) according to prediction in [9] will densify at the maximum rate (see Fig. 5a ). An increase in grain size with time during compact sintering is very slow (see Fig. 5b ). Results of theoretical analysis of sintering carried out above, have received numerous confirmations in experimental work. The authors of [10] studied sintering of submicron TiO 2 spherical powder. In one of the compacts particles packed with a high 69% density had dense particle packing with a narrow pore size distribution. Compacts with a lower 56% density had a wide pore size distribution. Comparison of kinetic compaction curves for these compacts during sintering and dependences for grain size on density achieved (Fig. 6) demonstrate the good conformity with calculated data [9].

4 Powder Compact Structure. Part In a model proposed in [9], no differences are given between the behavior during sintering of fine intra-agglomerate pores and coarse interagglomerate pores. In addition, the mechanism of pore shrinkage is markedly different. Densely packed intra-agglomerate areas of fine pores as a rule have a low coordination number and during development of sintering their boundaries are turned towards the center of a pore with a concave side (Fig. 7a ). In contrast to them, coarse interagglomerate pores have a high coordination number. The shape of their boundaries during sintering is turned towards the center of a pore with a convex side (Fig. 7b ). Studies have shown [11] that pore configuration has a strong effect on their behavior during sintering. At the start of sintering intra-agglomerate low-coordinated pores are in the vicinity of fine particles, and over whose numerous boundaries powerful mass transfer is accomplished into their inner cavity, and is maintained to complete compact densification. This environment in the initial sintering stage also exists for coarse interagglomerate pores. However, this environment does not survive to complete compact densification. Selective recrystallization in fine-grained intra-agglomerate areas with a developed boundary network, which is surrounded by pores, leads to total exhaustion of effective mass transfer path within them, and a reduction in coordination of interagglomerate pores up to their closure. Under these conditions weakening is observed for the sintering potential, or shrinkage slows down and proceeds on a background of grain growth. Grain growth in this stage starts to play a definitive role is structure formation. In their model the authors of [9] do not consider the difference noted above of behavior during sintering of different forms of pores (see Fig. 7), and are limited to analyzing sintering assuming dense particle packing, which is rarely realized in practice. Therefore an analytical method, developed for qualitative analysis of sintering [9] in spite of good conformity obtained in analyzing many experimental results, all the same does not provide reliable treatment of known factors observed during sintering of inhomogeneously packed powder compacts. In particular, the fact growth of high coordination interagglomerate pores. This phenomenon was analyzed in [12] for the first time from a position of physical sintering theory. The basis of the analysis is a geometrical model of a highly coordinated pores (Fig. 8). In this model a pore is represented by a space within a ring, comprising touching single size particles, i.e., grain with radius r. The radius of the circle described is associated with pore radius. Between grains in contact the internal space is occupied by ternary junctions, formed by intergranular and free boundaries, adjacent to a junction at contact angle. For this idealized configuration [2] clear conformity has been established between contact angle and the ratio of pore radii to particle radius, i.e., /r. Contact angle, equals zero in the original configuration, and it grows with pore shrinkage tending towards a value of an equilibrium dihedral angle. The magnitude of this angle for each material under specific conditions Fig. 6. Dependence of density on sintering time (a) and average grain size on achieved density (b ) for TiO 2 compacts with a different pore size distribution width [10]. Fig. 7. Pores: a) low-coordination between agglomerates; b ) high-coordination between agglomerates. (temperature, gas atmosphere) has a specific constant value, determined from surface force equilibrium conditions at a ternary junction. The condition for stoppage of pore shrinkage is the condition when contact angle reaches a value of equilibrium dihedral =. This equality clearly determines the value of ratio of pore and particle radii /r, with whose deviation equilibrium surface forces are disrupted and a pore acquires a potential for increasing its size or conversely for shrinkage. The value of the critical ratio /r depends on the number of particles surrounding a pore, i.e., its coordination number. With use of such a scheme [12] graphical representation has been plotted for the region of instability for high coordination pores. One of these diagrams is shown in Fig. 9. For example, it follows from it that for material with

5 452 A. V. Galakhov Fig. 8. Diagram of highly coordinated pore shrinkage during sintering: original pore on left; pore in position = on right. Fig. 10. Sintering of densely packed compact (L) and containing pores between agglomerates (H) of submicron TiO 2 powder [13]. Fig. 9. Fields of highly coordinated pore instability [12]. Fig. 11. Diagram of stability fields for submicron TiO 2 powder compacts with a different pore space structure [13]. = 120 an increase in pore coordination number more than 12 converts it to a series of growing pores during sintering, but pores with a coordination number less than 12 shrink in size during sintering. In fact, precise quantitative prediction for actual powder materials is difficult on the basis of these diagrams. Nonetheless, results obtained in many experimental studies confirm the scheme assumed in [12]. Results are given below, obtained in experimental studies of sintering powder compacts containing high coordination pores [13]. The authors of [13] worked with titanium dioxide powder with a narrow pore size distribution and submicron particle size of 0.24 m. Specimens for studying sintering were cast in an aqueous suspension on a gypsum mold. By regulating the aggregate stability within a suspension, two batches of powder compacts were prepared. Compacts of the first batch, prepared from a flocculated suspension, had low density, and as a result of this an extremely uniform loose structure containing coarse and interagglomerate pores, i.e., H (high). Compacts cast from a suspension with the optimum regulated aggregate stability, were dense and narrow, i.e., L (low). In order to study pore structure distribution during sintering a series of firings was carried out at 1200 C with different duration (Fig. 10). It is seen from Fig. 10a that the nature of the kinetic curves differs fundamentally. Whereas L compacts with a uniform dense structure reached a density close to 100% during a short time interval (1.5 h), for complete densification of an H compact containing highly coordinated pores the sintering time to a dense conditions reached 5 h. Whereas in L compacts with an increase in sintering time there was a continuous increase in density, in H compacts there is an intermediate time interval within which density remains unchanged. Only a marked increase in sintering time leads to gradual pore removal. The nature of change in pore structure during sintering is fundamentally different (see Fig. 10b ). In L compacts during the whole process there is a stable reduction in average pore size, and this disappears entirely after 30 min. In H compacts pores grow continuously and a tendency towards disappearance does not develop. This occurs on a background of an increase in material density. Results obtained [13] have been developed by the authors according to a procedure in [12], and presented as a diagram of pore instability, constructed for TiO 2 (Fig. 11). It is seen that an L with dense particle packing has pores

6 Powder Compact Structure. Part within the shrinkage region, whereas an H compact with coarse interagglomerate pores is in the pore growth zone. A thermodynamic approach formulated in [12] opens up new possibilities for analyzing sintering. For example, by means of this approach quantitative evaluation has been obtained for the limiting values of pore coordination number [15]. With an increase in coordination number above a limiting value pore shrinkage changes to growth. This evaluation was performed for a two-dimensional pore, formed by cylindrical particle of infinite length. Pores satisfying the relationship +2 /n < ( is equilibrium dihedral angle, n is pore coordination number), have internal boundaries concave towards the center (see Fig. 7a ), and acquire a potential for shrinkage. For pores with +2 /n the reduction in size disturbs the equilibrium condition, and therefore these pores grow during sintering. NUMERICAL METHODS An abundant number of physical and geometrical conditions, changing in each stage of powder compact sintering, point to the complexity of describing this process by a single analytical equation. This situation has pushed researchers operating in this field towards use of numerical methods. Moreover, implementation of them in view of the universal use of contemporary computer technology and accumulated volume of specialized software becomes simpler. Use of numerical methods for analyzing sintering has produced procedural approaches. It is possible to combine them into three groups: 1) numerical methods within the scope of a continuum model of sintering, 2) probability numerical methods (Monte-Carlo methods), 3) numerical methods within the scope of physical sintering model. Numerical methods within the scope of a continuum sintering model Implementation of a numerical method for analyzing sintering three stages: I) choice of geometric model of an object within which the process being analyzed proceeds; II) choice of a numerical discretization method for object shape; III) choice of a definitive equation on whose basis a process is analyzed. As a geometric model, used within the scope of a continuum sintering model, there is often use of a configuration of two cylindrical or spherical particles separated by a common boundary. For discretizing object shape (sintered particle) it is normal to use a finite element method [15 19]. As a definitive approach, describing change of a discrete continuum, a solid mechanics viscous flow equation is used 1 2, (6) where is strain date deviator; is stress deviator; is particle material viscosity. Fig. 12. Particle breakdown diagram with finite elements and neck shape evolution during sintering [15]. In fact, material viscosity depends on nature of material and temperature. Normal empirical values obtained by experiment are used. An example of breaking down a model of a sintered particle into finite elements and results obtained for of a neck-shaped contour evolution during sintering [15] are shown in Fig. 12. Use of solid mechanics methods is not limited to considering a two-particle configuration. This approach has been used for analyzing sintering of multiparticle powder molding [19]. A finite element method is used in order to describe shape change of powder particles. All of the analyzed continuums are broken down into finite elements, i.e., sub-regions, whose situation at boundaries obeys continuity conditions. Instead relationship (8) used normally in viscous flow theory, an original relationship containing porosity was proposed [19] as a definitive equation, P ( 3/ 2) /( 1 ) P, (7) eff where P eff is stress tensor deviator at a continuum point (all-round pressure); P is external pressure applied to a compact. In spite of the apparent generality of this approach the authors of the work were compelled to analyze the situation separately (different models), corresponding to the initial stages and the final stage (Fig. 13). In order to analyze the initial sintering stage a geometric of an object was used, represented by a discrete continuum of touching particles (see Fig. 13, upper row). In the concluding sintering stage the object analyzed is represented by a solid continuum, which contains isolated pores (see Fig. 13, lower row). Unfortunately, in this analysis there is no consideration of reaction of boundaries with cavities and pore growth in the concluding sintering stage. In other words, with use of a continuum model [15] it is impossible to follow structural changes in a sintering powder compact in all of the stages of material densification.

7 454 A. V. Galakhov Fig. 13. Finite element discretization and shape change is a particle assembly in the initial sintering stage and pore assembly in the concluding stage [15]. Fig. 14. Breakdown of model polycrystal region. Fields with orientation 0 represent pores [23]. Fig. 15. Grain growth and pore transformation within polycrystalline material in the concluding sintering stage. Modeling by the Monte-Carlo method [23]. Probability numerical methods Recently probability methods have acquired greater popularity in order to analyze sintering. By analogy with use of these methods in other areas they often have a generally accepted term, i.e., Monte-Carlo method. Initially this method was used for analyzing recrystallization and grain growth in polycrystalline materials [20, 21]. The model developed was modified in order to describe the concluding sintering stage in a polycrystal occurring on a background of rapid grain growth [22, 23]. The idea of analysis includes breaking down a model region of polycrystalline material by a regular micro-sub-region network whose crystal orientation determines the specific value of a discrete possible series. A diagram of a breakdown is shown in Fig. 14 [23]. Each crystal orientation is given is own index. A collection of neighboring micro-sub-regions with identical indices forms a continuous sub-region of considerable size, and this is treated as a single-crystal grain with a single orientation (in Fig. 14 intergranular boundaries are given between these sub-regions). One possible crystal orientation, designated by index 0, is fastened to vacant areas, i.e., pores. An expression is used to work out grain boundary energy of a system H J ( 1 ), (8) ij nn where J is boundary segment energy, separating neighboring microregions in contact; ij is Kronecker symbol with indices of neighboring sub-regions. Segment energy does not depend on the specific numbers of micro-sub-regions in contact. Its value is constant, i.e., the mutual orientation of particles in contact in figuring out boundary energy is considered according to the agree does not agree principle. For neighboring sub-regions with identical orientation (i = j) the local contribution of grain boundary energy of a system is zero, and only at a boundary where i does not equal j does it acquire a specific value. Equation (10) is used for working out grain boundary energy of a system with an approach by a regular network. A picture is shown in Fig. 15 of polycrystalline material structure transformation in the concluding sintering stage, obtained using the Monte-Carlo method [23]. It is seen that in this stage grain growth prevails strictly over sintering (removal of porosity). Recently the Monte-Carlo method has become generally accepted and is often used for analyzing the concluding stage of sintering and recrystallization in polycrystalline materials. Many questionable aspects are considered and corrected in recent editions of this program product [24]. In particular, drawing on a molecular dynamics method values of segment energy for boundary J in the basic Eq. (10) are clarified. The method may also be used in a three dimensional arrangement in production practice for optimizing heat treatment regime for specific objects [24]. The method is widespread in all sintering stages, including the initial, intermediate, and concluding stages [25]. In addition, the possibility is demonstrated of its use for analyzing sintering of multiparticle configurations. An original modification of this method is described in [26] that employs it for analyzing sintering combined with a finite element method. (To be continued)

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