علم سرامیک

Sintering and Microstructure of Ceramics 3.1. Sintering and microstructure of ceramics We saw in Chapter 1 that sintering is at the heart of ceramic processes. However, as sintering takes place only in the last of the three main stages of the process (powders → forming → heat treatments), one might be surprised to see that the place devoted to it in written works is much greater than that devoted to powder preparation and forming stages. This is perhaps because sintering involves scientific considerations more directly, whereas the other two stages often stress more technical observations – in the best possible meaning of the term, but with manufacturing secrets and industrial property aspects that are not compatible with the dissemination of knowledge. However, there is more: being the last of the three stages – even though it may be followed by various finishing treatments (rectification, decoration, deposit of surfacing coatings, etc.) – sintering often reveals defects caused during the preceding stages, which are generally optimized with respect to sintering, which perfects them – for example, the granularity of the powders directly impacts on the densification and grain growth, so therefore the success of the powder treatment is validated by the performances of the sintered part. Sintering allows the consolidation – the non-cohesive granular medium becomes a cohesive material – whilst organizing the microstructure (size and shape of the grains, rate and nature of the porosity, etc.). However, the microstructure determines to a large extent the performances of the material: all the more reason why sintering Chapter written by Philippe BOCH and Anne LERICHE. 56 Ceramic Materials deserves a thorough attention, and the reason for which this chapter interlaces “sintering” and “microstructures”. We will now describe the overall landscape and the various chapters in this volume will present, on a case-by-case basis, the specificities of the sintering of the materials they deal with. Sintering is the basic technique for the processing of ceramics, but other materials can also use it: metals, carbides bound by a metallic phase and other cermets, as well as natural materials, primarily snow and ice. Among the reference works on sintering, we recommend above all [BER 93] and [GER 96]; the latter refers to more than 6,000 articles and deals with both ceramics and metals. We also recommend [LEE 94], which discusses ceramic microstructures and [RIN 96], which focuses on powders. 3.2. Thermodynamics and kinetics: experimental aspects of sintering 3.2.1. Thermodynamics of sintering Sintering is the consolidation, under the effect of temperature, of a powdery agglomerate, a non-cohesive granular material (often called compact, even though its porosity is typically 40% and therefore its compactness is only 60%), with the particles of the starting powder “welding” with one another to create a mechanically cohesive solid, generally a polycrystal. The surface of a solid has a surplus energy (energy per unit area: γSV, where S is for “solid” and V is for “vapor”) due to the fact that the atoms here do not have the normal environment of the solid which would minimize the free enthalpy. In a polycrystal, the grains are separated by grain boundaries whose surplus energy (denoted γSS, or γGB, where SS is for “solid-solid” and GB for “grain boundary”) is due to the structural disorder of the boundary. In general, γSS < γSV, so a powder lowers its energy when it is sintered to yield a polycrystal: the thermodynamic engine of sintering is the reduction of system’s interfacial energies. Mechanical energy is the reduction of the system’s free enthalpy: ΔGT = ΔGVOL + ΔGGB + ΔGS where ΔGT is the total variation of G and where VOL, GB and S correspond to the variation of the terms associated respectively with the volume, the grain boundaries and the surface. Sintering and Microstructure of Ceramics 57 Starting particles Sintering without densification Sintering with densification and shrinkage Figure 3.1. Sintering of four powder particles. In general, we want sintering to be “densifying”, in which case the reduction of porosity implies a shrinkage: Lfinal= L0 – ΔL. Some mechanisms are non-densifying and allow only grain growth. This diagram shows a two-dimensional system but the powder is a three-dimensional system. We could consider an octahedral configuration where the interstice between the four particles is closed below and above by a fifth and a sixth particle [KIN 76] The interfacial energy has the form G = γA, where γ is the specific interface energy and A its surface area. The lowering of energy can therefore be achieved in three ways: i) by reducing the value of γ, ii) by reducing the interface area A, and iii) by combining these effects. The replacement of the solid-vapor surfaces by grain boundaries decreases γ, when γSS is lower than γSV. The reduction of A is achieved by grain growth: for example, the coalescence of n small spheres with surface s and volume v results in a large sphere with volume V = nv but with surface S < ns (this coalescence can be easily observed in water-oil emulsions). In fact, the term sintering includes four phenomena, which take place simultaneously and often compete with each other: – consolidation: development of necks that “weld” the particles to one another; – densification: reduction of the porosity, therefore overall contraction of the part (sintering shrinkage); – grain coarsening: coarsening of the particles and the grains; – physicochemical reactions: in the powder, then in the material under consolidation. 58 Ceramic Materials 3.2.2. Matter transport Sintering is possible only if the atoms can diffuse to form the necks that weld the particles with one another. The transport of matter can occur in vapor phase, in a liquid, by diffusion in a crystal, or through the viscous flow of a glass. Most mechanisms are activated thermally because the action of temperature is necessary to overcome the potential barrier between the initial state of higher energy (compacted powder) and the final state of lower energy (consolidated material). Atomic diffusion in ceramics is sufficiently rapid only at temperatures higher than 0.6-0.8 TF, where TF is the melting point (in K). For alumina, for example, which melts at around 2,320 K the sintering temperature chosen is generally around 1,900 K. 3.2.3. Experimental aspects of sintering The parameters available to us to regulate sintering and control the development of the microstructure are primarily the composition of the starting system and the sintering conditions: – composition of the system: i) chemical composition of the starting powders, ii) size and shape of the particles, and iii) compactness rate of the pressed powder; – sintering conditions: i) treatment temperature, ii) treatment duration, iii) treatment atmosphere and, as the case may be, iv) pressure during the heat treatment (for pressure sintering). Pressureless sintering and pressure sintering In general, sintering is achieved solely by heat treatment at high temperature, but in difficult cases it can be assisted by the application of an external pressure: – pressureless sintering: no external pressure during the heat treatment; – pressure sintering (under uniaxial load or isostatic pressure): application of an external pressure during the heat treatment. Pressure sintering requires a pressure device that withstands the high sintering temperatures, which is in fact a complex and expensive technique and therefore reserved for specific cases. Sintering with or without liquid phase Sintering excludes a complete melting of the material and can therefore occur without any liquid phase. However, it can be facilitated by the presence of a liquid phase, in a more or less abundant quantity. We can thus distinguish solid phase sintering on the one hand and sintering where a liquid phase is present; the latter Sintering and Microstructure of Ceramics 59 case can be either liquid phase sintering or vitrification, depending on the quantity of liquid (see Figure 3.2): – for solid phase sintering, the quantity of liquid is zero or is at least too low to be detected. Consolidation and elimination of the porosity require a disruption of the granular architecture: after the sintering, the grains of the polycrystal are generally much larger than the particles of the starting powder and their morphologies are also different. Solid phase sintering requires very fine particles (micrometric) and high treatment temperatures; it is reserved for demanding uses, for example, transparent alumina for public lamps; – for liquid phase sintering, the quantity of liquid formed is too low (a few vol.%) to fill the inter-particle porosities. However, the liquid contributes to the movements of matter, particularly thanks to phenomena of dissolution followed by reprecipitation. The partial dissolution of the particles modifies their morphology and can lead to the development of new phases. A number of technical ceramics (refractory materials, alumina for insulators, BaTiO3-based dielectrics) are sintered in liquid phase; – lastly, for vitrification, there is an abundant liquid phase (for example, 20 vol.%), resulting from the melting of some of the starting components or from products of the reaction between these components. This liquid fills the spaces between the non-molten particles and consolidation occurs primarily by the penetration of the liquid into the interstices due to capillary forces, then solidification during cooling, to give crystallized phases or amorphous glass. This type of sintering is the rule for silicate ceramics, for example, porcelains. However, the quantity of liquid must not be excessive, and its viscosity must not be too low, otherwise the object would collapse under its own weight and would lose the shape given to it. Sintering with and without reaction We can speak of reactive sintering for traditional ceramics, where the starting raw materials are mixtures of crushed minerals that react with one another during sintering. The presence of a liquid phase often favors the chemical reactions between the liquid and the solid grains. However, for solid phase sintering, reactive sintering is generally avoided: either we have the powders of the desired compound already, or sintering is preceded by calcination, i.e. a high temperature treatment of the starting raw materials to allow their reaction towards the desired compound, followed by the crushing of this compound to obtain the powders that will be sintered: – non-reactive sintering: an example is that of alumina, because the powders of this compound are available on the market; – calcination and then sintering: an example is barium titanate (BaTiO3). BaTiO3 powders are expensive and some industrialists prefer to start with a less expensive 60 Ceramic Materials mixture of barium carbonate BaCO3 and titanium oxide TiO2, the mixture being initially calcined by a high temperature treatment to form BaTiO3, which is then crushed to give the powder that will be used for sintering; – reactive sintering: an example is that of silicon nitride (Si3N4), for which one of the preparation methods consists of treating silicon powders in an atmosphere of nitrogen and hydrogen, so that the reaction that forms the nitride (3Si + 2N2 → Si3N4) is concomitant with its sintering (see Chapter 7). This technique (RBSN = reaction bonded silicon nitride) makes it possible to circumvent the difficulties of the direct sintering of Si3N4 and offers the advantage of minimizing dimensional variations, but the disadvantage of yielding a porous material (P > 10%). Mullite and zirconia mullite can also be prepared by reactive sintering [BOC 87 and 90]. Figure 3.2. Top, vitrification: the liquid phase is abundant enough to fill the interstices between the particles; in the middle, liquid phase sintering: the liquid is not sufficient to fill the interstices; bottom, solid phase sintering: organization and shape of the particles are extremely modified. This diagram does not show the grain coarsening: in fact, the grains of the sintered material are appreciably coarser than the starting particles [BRO 911] Densification: sintering shrinkage The starting compact has a porous volume (P) of about 40% of the total volume. However, for most applications, we want relatively non-porous, even dense, ceramics (P ≈ 0%). In the absence of reactions leading to an increase in the specific volume, densification must be accompanied by an overall contraction of the part: characterized by linear withdrawal (dl/l0), this contraction usually exceeds 10%. The control of the shrinkage is of vital importance for the industrialist: on the one hand, the shrinkage should not result in distortions of the shape and on the other hand, it must yield final dimensions as close as possible to the desired dimensions. In fact, an excessive shrinkage would make the part too small, which cannot be corrected, Sintering and Microstructure of Ceramics 61 and an insufficient shrinkage would make the part too large; in this case machining for achieving the desired dimension must be done by rectification, often by means of diamond grinding wheel – a finishing treatment all the more expensive as the volume of matter to be abraded is large. It is difficult to control shrinkage with a relative accuracy higher than 0.5%. Because of the phenomenon of shrinkage, dilatometry tests are widely used for the in situ follow-up of sintering: starting with the “green” compact to arrive at the fired product, a heating at constant speed typically comprises three stages: i) thermal expansion, accompanied by a vaporization of the starting water and a pyrolysis of the organic binders introduced to support the pressing of the powder; ii) a marked contraction, due to particle rearrangement, the development of sintering necks and granular changes; iii) a resumption of the thermal expansion of the sintered product. Many studies have sought to correlate the kinetics of shrinkage and the growth of inter-particle necks [BER 93, KUC 49]. Porosity is open as long as it is inter-connected and communicating: the material is then permeable to fluids. Porosity is closed when it is not inter-connected: even if it is not yet dense, the material can then be impermeable. The porosity level corresponding to the transformation of open pores to closed pores is about P ≈ 10%. Sintering generally occurs in the absence of external pressure applied during the heat treatments (pressureless sintering); the particles of the starting powders weld with one another to form a polycrystalline material, possibly with vitreous phases; the presence or absence of a liquid phase is important. Finally, the term sintering covers four phenomena: i) consolidation, ii) densification, iii) grain coarsening and iv) physicochemical reactions. The beginning of the densification is the usual sign for the beginning of the sintering, frequently followed by dilatometry experiments. 3.3. Interface effects From a macroscopic point of view, the driving force behind the sintering of a powder to form a polycrystalline material is the reduction of energy resulting from the reduction of solid-vapor surfaces in favor of the grain boundaries. The necessary condition for sintering is therefore that the grain boundary energy (γGB) is low compared to the energy (γSV) of the solid-vapor surfaces. But this condition is not always achieved, as shown by silicon carbide (SiC) or silicon nitride (Si3N4): materials where the γGB/γSV ratio is too high to allow easy sintering. The solution for sintering such materials can be i) the use of sintering additives chosen to increase 62 Ceramic Materials γSV or to decrease γGB or ii) the use of pressure sintering, which provides external work: dW = – PexternaldV. From a microscopic point of view, it is the differential pressure on either side of an interface that causes the matter transport making sintering possible. This pressure depends on the curvature of the surface. Interface energy The increase in energy (γ) at the level of the interfaces, due to the fact that the atoms do not have their normal environment, is always very insignificant: γ is typically a fraction of Joules per m-2. Substances added in very small quantities can have a marked effect – this is also the case with liquids, as shown in the use of surface active agents in washing powders and detergents. The surfaces of the particles and the grain boundaries of sintered materials are frequently covered by adsorbed species, segregations or precipitations, which means that interfacial energies are in general modified by these extrinsic effects. We can give the example of silicon-based non-oxide ceramics (SiC or Si3N4), whose particles are covered with an oxidized skin – silica. As the specific surface of a powder grows as the inverse of the squared linear dimensions of the grain, the interfacial effects are marked in fine powders, which is generally the case with ceramic powders – the diameter of the particles measures typically from a fraction of a micrometer to a few micrometers, which corresponds to specific surfaces in the order of a few m2g-1. The role of the curvature in the energy of an interface can be illustrated by considering a bubble blown in a soapy liquid using a straw. If we disregard the differences in density, and consequently the effects of gravity, the only obstacle for the expansion of the blown bubble under the pressure P is the increase of the energy at the interface. For a spherical bubble, the equilibrium radius r is the one for which the expansion work is equal to this increase in energy [KIN 76]: ΔPdv = γdA dv = 4πr2dr dA = 8πrdr ΔP = γdA/dv = γ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ r dr 􀊌rdr 4 2 8 π = 2γ/r We note that the difference in pressure ΔP is proportional to the interfacial energy (here γ = γLV, liquid-vapor interface) and inversely proportional to the radius of curvature. Sintering and Microstructure of Ceramics 63 For a non-spherical surface with main radii of curvature r´ and r´´: ΔP = γ ( ) '' 1 ' 1 r r + [3.1] Likewise, the rise h of a liquid in a capillary of radius r is such that: ΔP = 2γcosθ/r = ρgh where ρ is the density of the liquid and θ the liquid-solid wetting angle. The relation: γ = (rρgh)/(2cosθ) is used to assess γ by measuring θ. These capillary effects contribute to the vitrification of silicate ceramics, because the viscous liquid formed by the molten components infiltrates into the interstices between the non-molten particles. The difference in pressure through a curved surface implies an increase in vapor pressure and also in solubility, at a point of high curvature: ΩΔP = RT ln(P/P0) = Vγ(1/r´ + 1/r´´) where Ω is the molar volume, P the vapor pressure above the curved surface, and P0 the pressure above a plane surface. Thus: ln(P/P0) = (Ωγ/RT)(1/r´ + 1/r´´) = (Mγ/ρRT)(1/r´ + 1/r´´) [3.2] where R is the ideal gas constant, T the temperature, M the molar mass, and ρ the density. Equation [3.2] is the Thomson-Kelvin equation. For a spherical surface, this relation can be seen when we consider the transfer of a mole of the compound as a result of the vapor pressure on the surface, the work provided being equal to the product of the specific energy and the variation of surface area: RTlnP/P0 = γdA = γ 8πrdr As the variation of volume is dv = 4πr2dr, the variation in radius for the transfer of a mole is dr = Ω/(4πr2), consequently: lnP/P0 = (Ωγ/RT)(2/r), which is the above result when r´ = r´´ = r The sign convention is to consider that the radii of curvature r´ and r´´ to be positive for convex surfaces and negative for concave surfaces. Equation [3.1] 64 Ceramic Materials shows that ΔP = 0 for a plane surface (r´ and r´´ = ∞). A bump tends to level itself and a hole to fill itself. We can mention as an example the progressive restoration of the flatness in a skating rink surface striped by the skates, after the skaters leave the rink (it is primarily surface diffusion that makes ice get back a smooth surface). The concept of pressure on a surface is a macroscopic concept. At the microscopic scale, atomic diffusion in a crystallized phase occurs primarily due to the movements of the vacancies. However, the equilibrium concentration of the vacancies is less under a convex surface than under a plane surface, and it is higher under a concave surface than under a plane surface. Thus, the vacancies migrate from the high concentration areas to the low concentration areas, an action which implies a movement contrary to that of the atoms. The effects are all the more obvious according to how marked the curvature (1/r) is and therefore the smaller particles are: sintering is facilitated by the use of fine powders (diameters of about a micrometer). However, the pressure variations and the energies brought into play by the interfacial effects still remain very low. EXAMPLE 1.– for spherical alumina particles (γSVAl2O3 ≈1Jm-2), the surplus pressure associated with particles with a diameter of 1 micrometer is 0.2%. EXAMPLE 2.– at what size must an Al2O3 monocrystal be crushed to increase its energy from 500 kJmole-1 (500 kJ.mole-1 is a typical value of the energies brought into play in chemical reactions involving metallic oxides)? If γSVAl2O3 = 1 J m-2 and ρ0Al2O3 = 4.103 kg m-3, the answer is: at a size less than that of the crystal cell! EXAMPLE 3.– what is the energy variation when 1 kg of SiO2 powder composed of beads of 2 μm diameter sinters to give a dense sphere and without internal interfaces? If γSVSiO2 ≈ 0.3 J m-2 and ρ0SiO2 = 2.2. 103 kg m-3, the answer is: 20 kJ only. The development of a sintering neck is illustrated by the simple model of two isodiametric spherical particles (see Figure 3.3). The connection between the two particles is a neck in the shape of a horse saddle, with r´ < 0 depending on the concavity (in the plane of the figure) and r´´ > 0 depending on the convexity (in the plane tangent to the two spheres, perpendicular to the figure). The neck is a very curved area, which constitutes a source of matter towards which the atoms coming from the surface (the sintering is then non-densifying) or the volume (the sintering is then densifying) migrate. The movements of matter result in the progressive coarsening of the sintering neck and the consolidation of the material. Sintering and Microstructure of Ceramics 65 x Surface diffusion Liquid Radius of the sintering neck Figure 3.3. At the beginning of the sintering, the consolidation is done by evaporation of the surfaces and condensation on the neck (on the left); this mechanism is not densifying. If there is a liquid (on the right) the capillary pressure helps the penetration of the liquid in the interstice and the dissolution-reprecipitation effects contribute to the matter transport 3.4. Matter transport Even if the thermodynamic condition of sintering is met (ASVγSV > AGBγGB), for the process to occur, its speed must be sufficient. However, the matter transport in a solid is very slow compared to a liquid or a gas. This matter transport can come from an overall movement (viscous flow of vitreous phases or plastic deformation of a crystal), the repetition of unit processes on an atomic scale (atomic diffusion in a crystal), from transport in vapor phase (evaporation then condensation) or in liquid phase (dissolution then reprecipitation). Speed is significant only if the temperature is sufficiently high. The diffusion (D) in a crystal or the inverse of the viscosity (η) of glass vary as exp(E/RT), where E is the apparent activation energy of the process. The usual values of E are a few hundred kilojoules per mole. The normal sintering temperatures are about 0.6 to 0.8 TF, where TF is the melting point of the solid in question. The matter movement takes place from the high energy areas towards the low energy areas – primarily, the sintering neck between the particles. We must distinguish two cases depending on the location of the source of matter: – when the source of matter is the surface, the mechanism is non-densifying, which means that the spheres take an ellipsoidal form, without their centers approaching one another. There is no macroscopic shrinkage and the porosity of the granular compact is not reduced significantly. The decrease in interfacial energy primarily comes from the grain coarsening; – when the source of matter is inside the grains (near the boundaries, or near defects such as dislocations), the mechanism is densifying: there is shrinkage and reduction in porosity (see Table 3.1 and Figure 3.4). 66 Ceramic Materials For solid phase sintering, there are four ways of diffusion: i) surface diffusion, ii) volume diffusion (often called lattice diffusion), iii) vapor phase transport (evaporation-condensation), and iv) grain boundary diffusion: the boundaries are very disturbed areas, which allow “diffusion short-circuits”. For liquid phase sintering, we must add dissolution-reprecipitation effects or a vitreous flow. Finally, for pressure sintering the pressure exerted allows the plastic deformation of the crystallized phases and the viscous flow of the amorphous phases. Path in Figure 3.4 Diffusion path Source of matter Shaft of matter Result obtained 1 Surface diffusion Surface Sintering neck Grain coarsening 2 Volume diffusion Surface Sintering neck Grain coarsening 3 Evaporation- condensation Surface Sintering neck Grain coarsening 4 Grain boundary diffusion Grain boundaries Sintering neck Densifying sintering 5 Volume diffusion Grain boundaries Sintering neck Densifying sintering 6 Volume diffusion Defects, like dislocations Sintering neck Densifying sintering Table 3.1. Matter transport during a solid phase sintering [ASH 75] 3.4.1. Viscous flow of vitreous phases The difference in pressure on either side of a curved interface causes a stress σ (a stress has the dimension of a pressure) which causes a viscous flow ε of the glass. The flow rate dε/dt is proportional to the stress and inversely proportional to the viscosity η: dε/dt proportional to σ/η. In general, viscosity decreases exponentially when the temperature increases: η = ηO exp(Q/RT) → dε/dt proportional to σ/ηexp(Q/RT) where Q is the apparent activation energy of the process. Sintering and Microstructure of Ceramics 67 Figure 3.4. Matter transport during a solid phase sintering; mechanisms 1, 2 and 3 are nondensifying; mechanisms 4, 5 and 6 are densifying; ⊥ schematizes a dislocation [ASH 75] 3.4.2. Atomic diffusion in crystallized phases Fick’s first law J = –D(δc/δx) for a unidirectional diffusion along x [3.3] J is the flow of atoms passing through a unit surface, per time unit, D the diffusion coefficient of the species that diffuses and c its concentration. Fick’s second law (δc/δt) = D(δ2c/δx2) [3.4] Nernst-Einstein’s equation The “force” that acts on the atom that diffuses is the opposite of the chemical potential gradient. The mobility of the atom i is Bi, the quotient of the speed of the atom by the driving force: –Bi = vi /[(1/N)dμi/dx] [3.5] Ji = –(1/N)(dμi/x)Bici where N is the Avogadro number and μi the chemical potential of the species i. 68 Ceramic Materials Considering the activity equal to the unit: dμi = RTd(lnci) [3.6] Substituting [3.6] in [3.5] and comparing with [3.3], we obtain: Ji = –(RT/N)Bi (dci/dx) Di = kTBi, where k is the Boltzmann constant [3.7] Therefore, the diffusion coefficient is proportional to the atomic mobility. Besides, dμ/dx is proportional to the pressure gradient dP/dx: J ≈ (D/kT)(dP/dx) [3.8] The difference in pressure between the two sides of an interface causes a matter flow that is proportional to the pressure difference and to the diffusion coefficient of the mobile species [PHI 85]. NOTE.– D = D0exp(–Q/RT), so despite the presence of the term kT in the denominator of [3.8], it is the exponential of the numerator that is more important: an increase of T results in a rapid increase of J. NOTE.– the volume diffusion coefficient DV is expressed in m2 s-1 (or, often, in cm2s-1). As regards grain boundary diffusion (or surface diffusion), it is usual to consider the thickness of the grain boundary eGB (or the thickness of the superficial area eS), so that the diffusion term is written as DGBeGB (or DSeS), the coefficients DGB and DS being then expressed in m s-1 (or in cm s-1). 3.4.3. Grain size distribution: scale effects An essential objective in controlling the microstructure of a sintered material is to be able to control densification and grain growth separately. In a ceramic filter, for example, we want to preserve a notable porosity, with pores of sizes calibrated with respect to the medium to be filtered. In an optical porthole, on the contrary, we want the sintering to be accompanied by a complete densification (zero residual porosity) because the presence of residual pores would result in the diffusion of the light. However, we saw that certain matter transport mechanisms are non-densifying (like surface diffusion), while others are densifying (like grain boundary diffusion): the objective is to play on the sintering parameters in order to favor a particular mechanism. The size of the powder particles is one of the parameters at our disposal. Sintering and Microstructure of Ceramics 69 Although the powders in general consist of particles of irregular size and shape, the simplistic approach that considers isodiametric spherical particles makes a useful semi-quantitative analysis possible. The laws of scale [HER 50] specify the manner in which a phenomenon associated with a “cluster” of particles must be transposed in the case of a homothetic cluster p times larger. For isodiametric spheres, the laws of scale relate to the radius of the spheres (r). Thus, the time taken to obtain a certain degree of progress in a process depends on granulometry, according to a law of scale that varies with the process brought into play. If t1 is the time that corresponds to the small cluster and t2 the time that corresponds to the large cluster, then t1/t2 = (r1/r2)n = (1/p)n, where the value of n depends on the process. We are interested here in matter transport processes that ensure sintering. We will deal with only two cases (flow of a vitreous phase and diffusion-reprecipitation in a liquid) and will give the results for the other mechanisms. Viscous flow of a vitreous phase The flow rate dε/dt is inversely proportional to the viscosity η and proportional to the stress, whose form (see equation [3.1]) is γ/u, where u is the radius of the sintering neck. The duration δt of the transport of a given quantity of matter is inversely proportional to the speed, therefore: δtviscosity ∞ 1/(dε/dt) ∞ ηu/γ The radius of the neck u, is in a certain ratio k with the radius of the particles: u = kr. For a system that grows homothetically, particles p times coarser imply neck radii p times larger. Therefore, for this system that is p times larger: δtviscosity ∞ pηu/γ ∞ pηkr/γ [3.9] This result shows that the duration is proportional to the size r of the particles and therefore that the sintering time necessary to obtain a certain degree of consolidation varies inversely to the size of the particles; for example, dividing the size of the particles by ten reduces the duration of the sintering in the same ratio. Dissolution-reprecipitation in a liquid We suppose that the spherical particles are covered with a thin film of liquid, with a thickness of eL. The matter flow is: J ∞ (– DLiquid/ kT)(δP/L) where DLiquid = transport coefficient in the liquid. 70 Ceramic Materials The area of the section through which the flow of diffusion passes is A ∞ eLr, but the pressure at the points of contact between the particles is γ/u, a term that is proportional to γ/r, therefore: δP ∞ γ/u ∞ γ/r The volume of matter that must diffuse to make a given level of densification possible is proportional to the cube of the linear dimensions of the system and therefore proportional to r3. The time necessary to reach this level of densification is therefore: δtliquidphase ∞ (displaced volume )/(speeddiff x volumeatom) ∞ r3/(JAΩ) ∞ r3/[(Dliquid/kT)(γ/r2)eLRΩ] δtliquidphase ∞ [r4kT][Dliquid eLγΩ], therefore δtliquidphase proportional to r4 [3.10] The duration is proportional to the power of four of the size of the particles. Dividing the size of the particles by 10 helps, this time, to gain a factor of 10,000 in the sintering time. Through a similar reasoning, we can show that the grain boundary diffusion and surface diffusion make the duration vary to the power of four of the size of the particles, the volume diffusion to the power of three, and evaporation-condensation to the power of two. In short, liquid phase diffusion, surface diffusion and grain boundary diffusion (R-4 law in the three cases) are more sensitive to the reduction in size of the particles than to volume diffusion (R-3), evaporation condensation (R-2), and finally viscous flow (R-1). 3.5. Solid phase sintering 3.5.1. The three stages of sintering Solid phase sintering refers to the case where no liquid phase has been identified (but observations through electronic microscopy in transmission sometimes show the presence of a very small quantity of liquid phase, for example due to a Sintering and Microstructure of Ceramics 71 segregation of the impurities along the grain boundaries). Solid phase sintering takes place in three successive stages: – initial stage: the particle system is similar to a set of spheres in contact, between which the sintering necks develop. If X is the radius of the neck and R the radius of the particles, the growth of the ratio X/R in time t, for an isothermal sintering, takes the form: (X/R)n = Bt/Dm, where B is a characteristic parameter of the material and the exponents n and m vary according to the process brought into play. For example, n = 2 and m = 1 for viscous flow; n = 5 and m = 3 for volume diffusion; n = 6 and m = 4 for grain boundary diffusion; – intermediate stage: the system is schematized by a stacking of polyhedric grains intertwined at their common faces, with pores that form a canal system along the edges common to three grains, connected at the quadruple points (see Figure 3.5). The porosity is open. This diagram is valid as long as the densification does not exceed ≈ 90-92%, a threshold beyond which the interconnection of the porosity disappears; – final stage: the porosity is closed; only isolated pores remain, often located at the quadruple points between the grains (“triple points” on a two-dimensional section) but which can be trapped in intragranular position. Figure 3.5. Diagram of the porosity in the form of inter-connected canals along the edges of a polyhedron with 14 faces, typical of the intermediate stage of sintering [GER 96] 72 Ceramic Materials 3.5.2. Grain growth As the energy of the interfaces has the form γA, where γ is the specific energy of the interface and A is the surface area of the interface, the system’s energy can be reduced using two borderline cases: – pure densification: the particles preserve their original size, but the solid-gas interfaces (γSG) are replaced by grain boundaries (γSS), with a change in the shape of the particles; – coalescence and pure grain growth: the particles preserve their original form, but they change in size by coalescence, thus reducing the surface areas. Pure densification has never been observed: there is always a grain growth. Owing to the difference in pressure (ΔP ≈ γ/r), the atoms diffuse from the high pressure area towards the low pressure area. In addition, a curved boundary blocked at its ends tends to reduce its length while evolving to a line segment. Because of these two causes, the boundary moves towards its center of curvature. By considering (in two dimensions) triple points with angles of 120°, the grains with less than six sides have their boundaries with the concave side turned towards the inside: the evolution towards the center of curvature makes these small grains disappear. A contrary evolution affects the grains with many sides: the small grains disappear in favor of the coarser grains, which grow (see Figure 3.6). Figure 3.6. The pressure on the curved interfaces is such that the boundaries move towards their center of curvature: the small convex grains (less than 6 sides) disappear while the coarse concave grains (more than 6 sides) grow to the detriment of the neighboring grains; the grains with rectilinear boundaries have an appreciably hexagonal form [KIN 76] In normal grain growth, the average grain size increases regularly, without marked modification of the relative distribution of the size; the microstructure expands homothetically. This type of grain growth is the one observed in a successful sintering. Sintering and Microstructure of Ceramics 73 Secondary recrystallization (or abnormal growth, or discontinuous grain growth) makes a few grains grow rapidly, to the detriment of the more moderately sized grains. The final microstructure is very heterogenous, with coexistence of very coarse grains and very small grains. This type of microstructure rarely leads to favorable properties and therefore is generally avoided. In addition to the possibility of being homogenous or, on the contrary, heterogenous, the microstructure can be more or less isotropic. For the simple case of a mono-phased polycrystal, we can distinguish four cases: – equiaxed microstructure and random crystalline orientation of the grains (no orientation texture): the material is isotropic by effect of average; – equiaxed microstructure, but orientation texture: the matter loses its average isotropy and the overall anisotropy is all the more marked that the crystal in question is more anisotropic for the property concerned; – oriented microstructure, but no orientation texture: anisotropy; – oriented microstructure and orientation texture: maximum anisotropy. The polycrystal then offers properties close to those of the monocrystal – except for the intrinsic effect of the grain boundaries. We can cite as an example the case of graphite fibers (“carbon fibers”) used for the mechanical reinforcement of composites. The majority of ceramics are multiphased materials that comprise both crystallized and vitreous phases. Porcelain thus consists of silicate glass “reinforced” by acicular crystals of crystallized mullite, but we can also observe millimetric crystal agglomerates with a very porous microstructure (iron and steel refractory materials), or fine grained polycrystals (< 10 μm) without vitreous phases and with very low porosity (hip prosthesis in alumina or zirconia). It should be reiterated that, in addition to the chemical nature of the compound(s) in question, it is the microstructure of the material (size and shape of the grains, rate and type of porosity, distribution of the phases) that controls the properties. 3.5.3. Competition between consolidation and grain growth Densification – and therefore the elimination of the pores – occurs effectively only if the pores remain located on the grain boundaries (intergranular position), because then the matter movements can take advantage of the grain boundary diffusion. However, a too rapid grain growth – and therefore a migration of the boundaries – leads to a separation of the pores and the boundaries: the pores are then trapped in the intragranular position, where they are difficult to eliminate, because only volume diffusion remains active. If the objective is to sinter a material to its 74 Ceramic Materials ultimate density, and therefore eliminate all the pores, the growth of the grains must be limited. In addition to its role in the coupling between densification and grain growth, the size of the grains (Φ) of the sintered ceramics is, together with the porosity, the essential microstructural parameter. We can give five examples: – the brittle fracture of the ceramics is controlled by the size of the microscopic cracks, because the mechanical strength σf is proportional to Kcac -1/2 , where Kc is the toughness and ac the length of the critical microscopic crack. However, ac is of the same order as the size of the grain. This means that σf varies typically by Φ-1/2: ceramics with high mechanical strength (machine parts, cutting tools, hip prostheses, etc.) must be very fine-grained; – the particle composites based on partially stabilized zirconia use mechanical reinforcement mechanisms that depend on the size of the zirconia inclusions: if they are too small (< 0.3 μm) they remain tetragonal and if they are too large (1 μm) they destabilize towards the monoclinical form, with swelling and thus micro-cracking of the surrounding matrix. The optimal effect is achieved for particles of intermediate size, metastable tetragonal, which are transformed from tetragonal to monoclinical in the stress field of a crack that propagates itself; – the high temperature creep of refractory materials is often due to diffusion mechanisms: volume diffusion leads to the Nabarro-Herring creep and grain boundary diffusion to the Coble creep, with creep rates in Φ-2 and Φ-3, respectively: refractory materials must therefore be coarse-grained in order to slow down the creep; – ferroelectric or ferrimagnetic ceramics have performances sensitive to the size of the domains (size that interacts with the grain size) and to the migration of the walls (which is hampered by the grain boundaries): very fine grains are monodomain, and from them we have ferroelectric ceramics with very high dielectric constant or “hard” ferrimagnetic ceramics with very high coercitive field strength; – finally, the transport properties (electric conduction or thermal conduction) are sensitive to the intergranular barriers due to the structural disorder of the grain boundaries or to the presence of secondary phases that are segregated there: coarse grains mean fewer grain boundaries and therefore fewer barriers. 3.5.4. Normal grain growth In a fine-grained polycrystal heated to a sufficient temperature, the size of the grains grows and correlatively the number of grains decreases. The driving energy is the one that corresponds to the disappearance