Materials Science and Engineering A362 (2003) 81–105Processing techniques for functionally graded materials夽 B. Kieback a , A. Neubrand b,c,∗ , H. Riedel c a Institut für Werkstoffwissenschaft der TU Dresden, Mommsenstraße 13, 01062 Dresden, Germany b FB 11, Technische Universität Darmstadt, Petersenstr. 23, 64287 Darmstadt, Germany c Fraunhofer-Institut für Werkstoffmechanik, Wöhlerstr. 11, 79108 Freiburg, Germany Received 18 July 2002; received in revised form 7 February 2003 Abstract An overview of the achievements of the German priority program “Functionally Graded Materials (FGM)” in the field of processing techniques is given. Established powder processes and techniques involving metal melts are described, and recent developments in the field of graded polymer processing are considered. The importance of modeling of gradient formation, sintering and drying for the production of defect-free parts with predictable gradients in microstructure is discussed, and examples of a successful application of numerical simulations to the processing of functionally graded materials are given. © 2003 Elsevier B.V. All rights reserved. Keywords: Functionally graded materials; Powder metallurgy; Sintering; Melt processing; Process simulation 1. Introduction In a functionally graded material (FGM) the properties change gradually with position. The property gradient in the material is caused by a position-dependent chemical composition, microstructure or atomic order. In the case of a position-dependent chemical composition the gradient can be defined by the so-called transition function ci (x, y, z) which describes the concentration of the component ci as a function of position. Already in 1972, the usefulness of functionally graded composites with a graded structure was recognized in theoretical papers by Bever and Duwez [1], and Shen and Bever [2]. However, their work had only limited impact, probably due to a lack of suitable production methods for FGMs at that time. It took 15 more years until systematic research on manufacturing processes for functionally graded materials was carried out in the framework of a national research program on FGMs in Japan. Since then, a major part of the research on FGMs was dedicated to processing of these materials and 夽 This is an overview of parts of the key results obtained within the Priority Program “Functionally Graded Materials” funded by the German Research Foundation (DFG) in the years 1995–2002. ∗ Corresponding author. Tel.: +49-761-5142282; fax: +49-761-5142403. E-mail address:
[email protected] (A. Neubrand). 0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-5093(03)00578-1 a large variety of production methods has been developed [3–6]. The manufacturing process of a FGM can usually be divided in building the spatially inhomogeneous structure (“gradation”) and transformation of this structure into a bulk material (“consolidation”). Gradation processes can be classified into constitutive, homogenizing and segregating processes. Constitutive processes are based on a stepwise build-up of the graded structure from precursor materials or powders. Advances in automation technology during the last decades have rendered constitutive gradation processes technologically and economically viable. In homogenizing processes a sharp interface between two materials is converted into a gradient by material transport. Segregating processes start with a macroscopically homogeneous material which is converted into a graded material by material transport caused by an external field (for example a gravitational or electric field). Homogenizing and segregating processes produce continuous gradients, but have limitations concerning the types of gradients which can be produced. Usually drying and sintering or solidification follow the gradation step. These consolidation processes need to be adapted to FGMs: processing conditions should be chosen in such a way that the gradient is not destroyed or altered in an uncontrolled fashion. Attention also has to be paid to uneven shrinkage of FGMs during free sintering. Since the sintering behavior is influenced by porosity, particle size and shape 82 B. Kieback et al. / Materials Science and Engineering A362 (2003) 81–105 and composition of the powder mixture, these problems must be handled for each materials combination and type of gradient individually referring to the existing knowledge about the sintering mechanisms [7]. It has been demonstrated that the introduction of a controlled grain size gradient in addition to the composition gradient can balance different sintering rates [8]. In general, this optimization results in unjustifiable limitations for the gradients design. Additionally, variations in the particle packing density can lead to deformation of the part. To overcome some of these limits, the superposition of the internal driving forces for sintering with an external pressure by hot pressing [9–11] or hot isostatic pressing [12] is possible. If the temperature regions for densification are very different, application of a temperature gradient during consolidation [13], liquid-phase sintering [14], laser assisted sintering [15], and spark plasma sintering [16,17] have been proposed. Although many of these methods for functionally graded materials were already developed in the early 1990s, there were still limitations concerning material combinations, specimen geometry and cost in 1995. One of the goals of the German priority program “Functionally Graded Materials” was thus the improvement of existing and the development of new processing techniques for functionally graded materials. Besides the well established powder metallurgical techniques other production processes suitable for metals and polymers with a low melting point were investigated in the priority program. Particular emphasis was also placed on modeling of production processes. Process simulations may allow the prediction of suitable processing parameters for FGMs in the future and reduce the considerable amount of experimental effort which is still necessary to produce a graded material free of macrodefects. In the following sections on powder metallurgy, melt and polymer processing and modeling, the achievements of the priority program are described. Each field is introduced with basic background information and literature, but a comprehensive literature review of the international literature is not attempted. 2. Powder metallurgy The powder metallurgy (PM) or ceramic technology route for processing of materials and engineering parts includes powder production, powder processing, forming operations and sintering or pressure assisted hot consolidation. Powders of many metals, alloys, compounds and ceramic materials with particle sizes ranging from nanometers to several hundred micrometers are available from industrial sources or may be produced by the methods developed over decades in the field of PM [18] or ceramics. Since the material used to form the sintered part is dispersed into very small portions of individual powder particles nearly ideal conditions exist to build-up graded materials with varying chemistry or microstructure by using the particles as building blocks. Practical considerations towards an efficient process design suggest the use of powder mixtures with changing average particle size or composition during deposition of the material prior to the forming operation. It depends on the method applied for powder deposition whether a smooth change or a stepwise variation is obtained in the green body. Since the consolidation of the green parts during sintering or hot pressing requires high temperatures at which diffusion processes are fast enough to enable densification, chemical reactions between particles of different compositions and the influence of particle size on the sintering behavior [7] have to be taken into account for the final microstructure and dimensional control of the part. On the other hand thermodynamic factors acting during sintering may be used to create a gradient, e.g. during liquid-phase sintering. By the powder metallurgy route the following types of gradients can be processed: (i) Porosity and pore-size gradients: Porosity gradients may be created either by the deposition of powder mixtures with different particle shape or by varying the deposition parameters including the use of space holders. Pore-size gradients in general are produced by the variation of particle size. The different sintering behavior of the powders requires special attention to manage the problems of part distortion but may also be used to produce a porosity gradient. (ii) Gradients in chemical composition of single phase materials: The deposition of powders with a continuous change in the composition of the mixture during sintering will lead to a single phase material with a smooth change in the distribution of elements if the phase diagram shows solubility over the chosen range of composition. The condition that has to be fulfilled is that the diffusion length of the different atoms is bigger than the average particle size or the characteristic width of the layered structure if stepwise deposition is applied. The starting powders may be elemental or chosen in the range of compositions between which the gradient should be formed. No application has been reported for a single phase material so far, but gradients in the microstructure and properties after cooling and heat treatment caused by small changes in the elemental composition might be interesting for intermetallic phase materials based on gamma titanium aluminide. The amount of ␣2 and ␥-phase in these two phase materials and the type of microstructure (lamellar, globular or duplex) obtained after thermal treatment depend directly on the Ti/Al ratio. (iii) Gradients of the volume content of phases and grain size gradients in two or multiphase materials: Most of the published literature focuses on gradients of the volume content of different phases. The phase diagram of such a multi-component system is characterized by no or limited solubility. In the latter case grain growth of the soluble phase during the heat treatment has to be taken into account, in particular if a liquid 2. Powder suspensions suitable for deposition with an air brush system were originally developed to deposit thin powder layers on a substrate [41].1.1. 1) was developed for this purpose [48. Slurry dipping and slip casting. The techniques to obtain a graded material by the powder metallurgy route [4. If a porous body is sequentially dipped into slurries with varying powder characteristics. During cooling the mixture solidifies and a free-standing body is formed. .1. 2.1. Complex FGM parts can be varied by stamping or laser-cutting the sheets to different sizes [39]. Wet powder spraying. By including a mixing system and controlled feeding of two or more suspensions graded powder layers can be deposited on a flat. liquid drag into pores by capillary forces leaves surface layers with a stepped gradient behind [45]. The minimum thickness of the layers is controlled by the size of the sprayed droplets and may be less than 50 m. A CAD model of a part was sliced into individual layers and a defined composition was associated with each layer.43]. 2.3.B. 2. A German research group used wet powder spraying for the manufacture of pore-size gradient microfilters [44].1. The disadvantages of the process are obvious: discrete changes. By changing the feedstock composition for Fig. In a project of the priority program a two piston construction with constant volume flow combined with a static mixing chamber with a very small mixing volume for a volume flow rate of 32 mm3 /s (Fig.1.1. Therefore a special synchronized distributor was designed which makes a continuous change of the composition feasible.38].1. This class of computercontrolled. The combinations of phases can include metal–metal. Sequential deposition of powders results in a stepped gradient.25–29] are classified in the following section according to their capability to produce a stepwise or continuous change of the composition of the powder compact. For powder rolling only powders which result in good green strength are suitable.1. limited number of layers (up to 10 in laboratory scale.1. tool-less manufacturing methods can be readily adapted to the production of FGMs.40]. Solid freeform processes. Microstructural gradients can also be produced by controlling the phase equilibrium during liquid-phase sintering with gradients of the components [19–24].1. Continuous dry deposition of layers. In the jet solidification process a hot powder-binder mixture with suitable flow properties is deposited with an extrusion jet moved in two dimensions.6.47]. Deposition of layers of powder mixtures with stepwise changes in the mixture 2. In some studies [39. 2. The same principles are valid for sequential slip casting [46. The high productivity of the green sheet production offers a high potential for up-scaling by laminating sheets with different compositions. curved or rotating substrate [42. A continuous process for the deposition of the powder mixture with changing compositions on a conveyor belt was developed in [36]. Hot press- 83 ing is used to join the layers during the final consolidation step. Nevertheless this method allows effective laboratory studies of functionally graded systems.1. Simultaneous drying allows building up parts reaching millimeter thickness with only small variations from layer to layer. If the powder composition is varied from layer to layer three dimensional graded bodies can be formed.1. Powder rolling is a well developed process giving green sheets with a thickness in the range of 1 mm. limited size of the part (<100 cm2 ) due the limits of compaction forces. 2. These methods have a high potential for series production if a limited number of layers fulfils the needs of the application. while tape casting needs a binder that has to be removed before sintering. 2. The resulting microstructure consists of two or more phases with gradients in volume content or in grain size which were predetermined during the deposition of the powders.1. The number of sheets would be limited mainly by the costs of fabrication. Two piston construction for the multiphase jet solidification equipment. Die compaction of layers (powder stacking). Thin sheets with different compositions can be produced by dry or wet powder techniques (powder rolling [18] or tape casting) and joined to form a stepped gradient [37.49]. / Materials Science and Engineering A362 (2003) 81–105 phase is used for activated sintering. metal–ceramic and ceramic–ceramic systems. discontinuous manufacturing with low productivity. Sheet lamination. but not more than two or three in potential fabrication).4. Tape casting of very fine powders even allows a sheet thickness in the double-digit micrometer range. Formation of graded powder compacts 2. this step was accompanied by a simultaneous SHS-reaction. limited thickness of individual layers (normally not less than 1 mm).5. In this simple and well established method a gradient is formed by the deposition of powder layers with changing compositions in the compacting die [30–35]. Kieback et al. 1.1.1. As a result. 2. Centrifugal sedimentation was used in one project of the priority program aiming at the development of ceramic filtration elements with a pore-size gradient. a pore size or composition gradient is produced.1. If the sprayed material contains a component that is not removed during the burnout process. A related process is impeller-dry blending where dry powders are mixed and homogenized in an impeller chamber prior to sedimentation in a mold [56]. Other methods. and the process is restricted to flat FGM parts. The velocity of ceramic particles in suspensions ν depends on the electrophoretic mobility µ and the electric field strength E. The formation of a graded structure by de-mixing of a powder suspension can be achieved for finer particles using centrifugation [61]. Due to the limited concentration in the suspension only thin layers can be produced.1. / Materials Science and Engineering A362 (2003) 81–105 each layer gradients were produced. The method is limited to cylindrical parts but offers a great flexibility in gradient design which is independent of the powder characteristics [53. Another obvious disadvantage of sedimentation processes are limitations of the types of gradients that can be produced. 40 mm height and up to 8 mm wall thickness. A tube like geometry may be produced by moving the form or the distributor plate along the axis of rotation during deposition. For smaller parts and thinner individual layers the three-dimensional printing process is more appropriate. Gravity sedimentation. pressure filtration. each layer of the part is created by spreading a thin layer of powders and selectively joining the powder by ink-jet printing a binder material. SiC–TiC was chosen as a non-reacting system. problems due to interparticle interactions occur frequently. 2. 2. microstructural gradients can be manufactured by 3D printing [50.2. A green body of sufficient strength is formed by simultaneously spraying an organic binder onto the wall. ν = µE (1) The mobility is defined by the properties of the dispersant and the zeta potential of the particles ζ: ε0 εr µ= ζ (2) η where η is the viscosity and ε0 εr the dielectric constant of the dispersant.54]. The laboratory equipment used in [58] was designed for small cylindrical FGM parts of 50 mm diameter.84 B. 2.2. No binder is required.4. Electrolytic decomposition of the aqueous electrolyte and the resulting formation of gas bubbles in the deposited layer were prevented by deposition of the particles on .1.% suspension containing Al2 O3 and ZrO2 .1. Electrophoretic deposition from suspensions containing more than one component can be used to produce graded bodies.51]. After repeated operations the unbound loose powder is removed to reveal a 3D green body. The viscosity of the feedstock (solid content: 47 vol. The bottom layer will still have the average composition of the suspension followed by an increase of de-mixing and the top of the sediment contains only the powder fraction with the lowest sedimentation velocity. Centrifugal powder forming in combination with liquid-phase sintering was used in the priority program for the production of W/Cu FGMs [55] with a ring diameter of 120 mm. In spite of its limitations this method has a potential for the manufacture of tile-type FGM products with bigger dimensions due to its simplicity and reproducibility. powder mixtures with a continuous and computer-controlled change of the composition are fed onto a rotating distributor plate which accelerates them towards the inner wall of a rotating cylinder. Gravity or centrifugal sedimentation. but have a bigger potential since they can be applied to achieve continuous changes in the material. Centrifugal sedimentation. The laser sintering process combines the deposition of powder layers and subsequent sintering [52] and can be modified into a stepped gradient processing method. If the sediment is directly formed from the suspension a complicated transition function is retained.7. Kieback et al. In the simplest case an external mixing system supplies suspensions with the variable concentrations of the components or the second component is added with time in calculated proportions [64–67].5% were obtained regardless of the SiC:TiC ratio. If sedimentation occurs in a liquid column free of particles a gradient with a continuous increase or decrease of the concentration of one particle type will be formed. With 8–10% liquid-phase sintering additives final densities of 97–99. During sedimentation in a column differences in the particle velocity caused by different density or size of the powder particles lead to de-mixing of the different particle types [57–59]. If sedimentation is carried out in highly loaded suspensions.1 and 1 mm were made by centrifugal deposition of TiO2 powders from aqueous suspensions containing a mixture of particle sizes between 10 nm and 5 m [62. In CPF.2. electrophoretic deposition.1. Deposition of powders with continuous changes in the mixture 2.2. 2. The feasibility to produce a graded green body from homogeneous suspensions using the difference in the electrophoretic mobility of different powders was investigated in the course of a priority program project [68]. In this process.%) and the sintering behavior had to be adjusted for varying compositions.2.3.1. A similar gradient will be obtained for a filtration process accompanied by sedimentation [60]. Centrifugal powder forming (CPF) and impeller-dry blending.1.2. Electrophoretic deposition.1. The sample height after the final consolidation was 10 mm or more. Pore-size graded filters of 18 mm diameter and a thickness between 0. In a project of the priority program [68] the higher mobility of Al2 O3 compared to ZrO2 was used to prepare a gradient from a 2 vol.63]. and centrifugal powder deposition may be used for the stepped type of gradient structures. 70]. 2. however a direct transition from porous to dense SiC leads to operating stresses exceeding the strength of the SiC.B. Pressure filtration/vacuum slip casting. In solid state sintering of bulk specimens diffusion lengths are usually small Fig. After sintering. Accordingly.2. By continuously changing the powder composition supplied to the filtration system it is easy to obtain a defined one dimensional gradient in the deposit [69]. 3).5. The aim of one priority program project was the development of a SiC fuel evaporator tube for a gas turbine combustor. Due to the higher mobility of Al2 O3 . 2a) and one using liquid pressure (Fig. 2b) were tested. By using segmented filter plates that allow for a local adjustment of the filtrate flow the production of axial and radial gradients is possible [71. If the surface of the evaporator tube is porous a dramatic increase of the evapo- 85 ration rate can be achieved. The same principles can be applied to slip casting [47. A variation of space holder concentration from 0 to 40 vol. Diluted suspensions were essential to prevent agglomeration and collective deposition due to particle interactions at higher concentrations. the gradient of the zirconia concentration was accompanied by a grain size gradient because zirconia inhibits the grain growth of Al2 O3 . Kieback et al. After sintering the porosity in the tube was graded from 10 to 50% over a distance of several millimeters (Fig. the zirconia concentration increased with time.1. (b) using controlled liquid pressure. Design of the filtration apparatus (a) using controlled gas pressure. A proper choice of the surfactants and pH was necessary to guarantee stability of the suspension and to achieve differences in electrophoretic mobility which were sufficient for de-mixing. FEM calculations [60] showed that a properly designed porosity gradient leads to a decrease of the failure probability at operating conditions by a factor of 106 compared to a non-graded tube. Despite the resulting longer deposition times graded deposits with several mm thickness could be produced.2. 2.% could be achieved. 2. Sintering The gradient that has been formed in the powder compact needs to be preserved during sintering. / Materials Science and Engineering A362 (2003) 81–105 a porous membrane. Pressure filtration with a variable concentration of space holder particles was selected to fabricate the required porosity gradients.72]. . one using gas pressure (Fig. if the permeability and evaporation rate are kept constant. it deposited faster and its concentration in the suspension and deposit decreased with time. Two experimental set-ups. Methods not requiring a sintering step Coating techniques using powders for the materials supply easily can be adopted to gradient fabrication by including a mixing unit in the powder supply chain (e. and that a surface layer of up to 60 m thickness enriched or depleted of binder metal could be adjusted. Ti. A continuous change of W.24]. [77. / Materials Science and Engineering A362 (2003) 81–105 Fig. In hard metals the interaction of the gas atmosphere with the compact during sintering produces chemical gradients with a strong influence on the phase content in a near-surface layer. A detailed study performed in the priority program project showed a very complex behavior for the gradient formation [20].78]). it has been demonstrated by Colin et al. If the particle size of the solid component was graded. It was found that the cooling regime after liquid-phase sin- 18 16 14 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Distance [mm] Fig. On the other hand. it is possible to make conscious use of liquid flow during liquid-phase sintering for gradient formation. Distortions of the parts can be prevented by carefully adjusted green densities of the different regions. Two V24/8: WC0.3. The same effect was found if the proportion of cubic and hexagonal carbides is graded or if the carbon or nitrogen content varied in the green compact [20]. The gradient formation starts with melting of the liquid phase and is fast. 2. During liquid-phase sintering. Feeding of mixed powders with controlled compositions is also the basis of the laser cladding technique for building up graded coatings or graded freeform components [80–87]. Nevertheless. 4. and a gradient in the liquid phase is easily destroyed. macroscopic flow of the liquid phase can easily occur. and Co or Ni content was obtained by slow diffusion controlled processes if carbon or nitrogen changes were produced by interactions with the sintering atmosphere.6µm6Co WC11µm15Co 20 Co-Content/Hardness/Fracture Toughness enough to fulfil this condition. Kieback et al. These redistribution processes are driven by the tendency of the system to minimize the free energy of internal surfaces and interfaces. however. [73] that it is possible to preserve concentration gradients during liquid-phase sintering of WC/Co hard metals. Another possibility is the use of two spraying guns with defined deposition rate and motion of the guns (e. Gradients in binder content and hardness can extend up to several millimeters (Fig. Co-concentration (䉫). this was studied extensively. Porosity gradient in a sintered ceramic sample produced by pressure filtration with PMMA beads as space holders. liquid redistribution by capillary forces occurred and produced a concentration gradient [20]. [79]). Zr.g. Lines are guides for the eye only.86 B. hardness (䊐) and fracture toughness () gradient in a hard metals with a gradient of the carbide grain size. 4). 3. In WC/Co hard metals a carbon gradient obtained by decarburizing results in a Co-rich graded surface layer [23. . In a priority program project.g. Tough surface layers free of mixed carbides for steel cutting tools have been obtained and studied [74–76] and the role of nitrogen was described [22]. tering is very important. 5. Kieback et al. For this study NiBSi. particles of a refractory phase are dispersed in a metal melt. Sedimentation casting 550 In the framework of the priority program. 6).% Cr3 C2 . [89]) than the melt have been used to prepare functionally graded aluminum tubes which are selectively particle reinforced at the inner or outer surface of the tube. Particles with a lower density (Mg2 Si. Laser and process parameters have to be optimized to obtain crack and pore free samples and to control the microstructure. Right: graded NiCrBSi tube with 0–80 vol. gradient formation by sedimentation of particles was investigated [91]. 5). These particles may be formed in situ during cooling of the melt [88] or dispersed in a preceding step.2. The main difficulties to produce graded parts by powder technology arise from the consolidation of the inhomogeneous powder compact during sintering.1. Melt processing Gradient formation can be conveniently achieved by transport processes in the molten state and subsequent consolidation. 6. In conclusion the disperse state of the materials in powder metallurgy and ceramics processing offers a great variety of methods to obtain a graded microstructure or composition in the final part. Some of the methods described are very cost-effective and have a good potential for industrial application. NiCrBSi and NiCr50 alloys and fine (10–45 m) and coarse (45–106 m) Cr3 C2 powder were tested. 3. The impedance gradient material was produced by dispersing WC particles in a Cu/Mn alloy and subsequent 450 350 250 0 10 20 30 40 50 60 Carbide Content [vol-%] 70 80 90 Fig. 3. Others will be reserved for laboratory studies or special applications. Most methods create a one dimensional gradient. The goal of the project was the production of a one dimensional FGM with an acoustic impedance gradient (the acoustic impedance is defined as the product of density and sound velocity).3 750 87 avoided and a thick wear resistant graded coating can be manufactured by laser cladding. FEM calculations demonstrated that stress concentrations at the interface with the carbide free substrate are 950 NiBSi + Cr3C2 NiCrBSi + Cr3C2 NiCr50 + Cr3C2 850 Hardness HV0. / Materials Science and Engineering A362 (2003) 81–105 Fig. Impedance graded FGMs can reduce the acoustic mismatch between a high impedance PZT ultrasound transducer and a low impedance material such as body tissue. [88]) and a higher density (Al3 Ti. NiCrBSi + Cr3 C2 and NiCr50 + Cr3 C2 cylinders obtained by laser beam cladding. Melt processing is widespread for FGMs containing a metal as one constituent. Hardness profile of graded NiBSi + Cr3 C2 . If platelets are used as particles a gradient in particle orientation can be produced during the casting process [90].B. Centrifugal casting In centrifugal casting. As an example the hardness profile of a cylindrical part with wide change of the carbide content (fine grade) is shown in (Fig. 650 3. Each graded system requires carefully selected powders and sintering conditions or the application of external pressure to reach full density without distortion. separately controlled pneumatic powder hoppers were used to feed Ni-alloy and Cr3 C2 powder simultaneously to deposit tracks with a height 100–400 m and a width 400–800 m with a 600 W Nd-YAG laser (Fig. . The carbide content was increased from 0 to 80 vol. but developments in free form manufacturing and special deposition units open up the three dimensional design of gradients. The density difference between particles and the melt leads to the formation of a particle concentration gradient if the melt is cast in a centrifuge.%. Subsequent cladding of overlapping tracks and deposition of tracks onto each other with a changing ratio of the powder feed rates produces a graded freeform component. Other methods using powders as starting materials or PM-related techniques that can be modified for gradient processing are the Osprey-Process and centrifugal casting with a SHS-reaction. Left: generation of a metal–carbide composite with compositional gradient by laser beam cladding. (b) specimen with ungraded interface. .3. graded and un- Fig. In a thermal shock experiment. and the width of the graded interface was predominantly controlled by the degree of solidification of the first melt at the time when the second melt was cast. In contrast. Controlled mold filling In a further project of the priority program. and in a further step. the second melt was cast on the partially solidified first material (Fig. In a gravity casting process. and components with a cylindrical shape could be obtained. After 700 thermal cycles a graded specimen showed no damage (Fig. 9. 7). A356/Duralcan composites after 700 thermal cycles at 400 ◦ C: (a) specimen with graded interface. 8. Gradient formation occurred by forced and thermal convection. 8). 7. an ungraded specimen displayed cracks already after 300 cycles which eventually lead to a complete destruction of the specimen (Fig.88 B. 9a). 9b). graded A356/Duralcan composites were heated to 400 ◦ C and quenched in water. the mold was partially filled with the first melt. Efforts to produce W/Al and W/Sn FGMs with this method failed due to solubility problems (W/Al) or poor wetting of the particles with the melt (W/Sn). Fig. In a modification of this process casting was carried out in a rotating mold. In this way a particle concentration gradient from 0 to 60% WC was produced (Fig. Kieback et al. The high impedance WC side matched the impedance of PZT exactly. Multi-alloy gravity casting process. 3. sedimentation. The controlled casting process was used to produce AlSi7 /AlSi18 and A356/Duralcan graded materials. Micrograph of a WC/Cu/Mn alloy gradient and the corresponding spatial distribution of the WC particle concentration. the functionally graded material was produced by successive casting of two melts [92]. / Materials Science and Engineering A362 (2003) 81–105 Fig. 10b). This can be achieved by natural convection in a vertical resistance furnace (Fig. (䉬) 264 m/s velocity in an induction furnace. a solutally and thermally unstable density gradient in the melt was generated. The degree of segregation in the induction furnace decreased with increasing sample velocity and segregation-free microstructures were generated at velocities above 0. For alloying elements with similar density as the matrix element. The minimum concentration at a velocity of 0. For this velocity.%Cu alloys.B.%Cu alloy even at the lowest sample velocity of 0. A temperature gradient was generated by inserting cooling coils in the top end of the furnace. The solidification direction was thus upwards. onedimensional concentration gradient with radial symmetry by directional solidification were also investigated in the framework of the priority program. Samples were melted inside the furnace and then moved toward the chilled zone with a constant velocity. 3. Set-up for directional solidification experiments with natural convection (a) and forced convection (b).%.3 mm/s (Fig. A large temperature gradient was established by submerging the tip of the sample in cooling water causing forced convection in the melt (Fig. Using natural convection.0 m/s. Concentration distributions in Al-4 wt. 11). 10a). The main reason for the higher stirring efficiency in the induction furnace was the radial temperature gradient introduced by the preferential heating of the sample surface. 11.% for complete mixing. Better mixing could be realized with forced convection in the induction furnace for specimens with 8 mm diameter (Fig. solidified with (䊏) 0. the difference between the solidus and liquidus composition during directional solidification of an alloy is used to prepare a FGM. / Materials Science and Engineering A362 (2003) 81–105 89 Fig. 10. 11).4. and for alloying elements with higher density than the matrix element.34 m/s. Directional solidification The possibilities of generating a macroscopic. In order to avoid solidification under steady state with constant solid concentration. Kieback et al. the melt in front of the solidification front must be stirred as effectively as possible. In this process. .9 m/s. an induction furnace was used. a Scheil type concentration profile with the maximum possible degree of segregation was obtained.56 wt. which is close to the theoretical value of 0. Fig. (䉱) 46 m/s. complete mixing could not be achieved during directional solidification of an Al-4 wt. (䊉) 9.48 wt.9 m/s was 0. 2 MPa is necessary for infiltration. even for a wetting angle of about 30◦ a pressure of about 1 atm is necessary for complete infiltration of a particulate preform [93]. Sketch of the experimental configuration for electrochemical gradation. Since this is accompanied by distortion.55]. and solid freeform fabrication [100]. coating of a polymer foam with ceramic slip in a centrifuge and burn-out of the polymer [99]. The preform must contain only open pores and must be insoluble in the melt. by suitably alloying the melt or changing the composition of the gas atmosphere. For a cylindrical pore in a ceramic/metal system with a pore radius of 5 m. The experimental set-up consists of a simple electrolysis cell with a porous anode material that undergoes dissolution (Fig. Further limitations of infiltration processing are caused by the restricted availability of graded porous preforms which must not contain closed porosity and need to have a sufficient mechanical strength at high porosity in order to resist the pressure applied during infiltration. During sintering. the particle size gradient leads to the formation of a porosity gradient [94]. W/Cu gradient formation: (a) starting condition after centrifugal forming.90 B. a wetting angle of 120◦ and a surface tension of 1 J/m2 a pressure of 0.5. Infiltration processing Infiltration is a suitable processing method for functionally graded materials containing phases of very different melting points. porosity gradient in the W/Cu composite [54. transformation of a Al/Al2 O3 gradient into a porosity gradient during sintering [96]. However. 13. . Larger pores can be infiltrated by applying pressure during infiltration. Kieback et al. An alternative to pressure application is the improvement of the wettability by precoating the preform. Infiltration of the pre-sintered body with additional Cu leads to a fully dense graded W/Cu composite (Fig. One processing method for such preforms is based on powder compacts with a graded particle size. 13). Infiltration processing is particularly attractive for metal/ceramic and glass/ceramic FGMs. (b) porosity gradient after liquid-phase sintering. 12). metal/ceramic systems are generally non-wetting (with a wetting angle Θ > 90◦ ) and the preforms are not infiltrated spontaneously. In a project of the priority program a combination of centrifugal powder forming and liquid-phase sintering was used to prepare a porosity gradient. and the movement of the liquid Cu towards a more homogeneous distribution caused a Fig. potential gradients inside the porous anode develop and lead to a local variation in the porous anode contact electrolyte cathode contact y z 0 x L Fig. (c) W/Cu gradient after Cu-infiltration. transformation of a gradient in carbon content in TiCx into a porosity gradient during sintering [97]. However. shape stabilization during sintering is required. / Materials Science and Engineering A362 (2003) 81–105 3. plasma spraying with varying process parameters during coating deposition [98]. but can be reached using squeeze casting or gas-pressure sintering equipment. For this purpose the amount of liquid forming Cu was graded in a W/Cu green body by centrifugal powder forming. Electrochemical dissolution of porous precursor materials as a route to produce graded preforms has been investigated in a different project of the priority program. Other methods which have been used to produce graded preforms include sintering of green bodies with a graded content of short fibres [95]. Pressures required for complete infiltration of real pore structures are higher. The required pressure p for a complete infiltration of a cylindrical pore of radius rp is given by p= −2γLV cos Θ rp (3) where γ LV is the surface tension of the melt. 12. In this process a preform of the more refractory phase possessing a porosity gradient is produced and infiltrated with the melt of the lower melting component at elevated temperatures. If a current passes through the cell. After water removal by filtration and drying.107]. Kieback et al. the carbon was burnt out [104]. [105]. The principle of this method is to introduce ceramic particles into the fiber bundle thereby acting as spacers. It was found that the suspension concentration and the winding speed of the fibers are the most important parameters to determine the fiber volume content of the preform (Fig. Porosity graded tungsten preforms also have been produced by this process [103]. The retained porosity graded ceramic preform can be infiltrated with a metal to produce a metal/ceramic FGM. (5) graded Al2 O3 preform after sintering. The retained ceramic preform was infiltrated with Al to produce a Al/Al2 O3 functionally graded material. The compacted foams are infiltrated with a ceramic suspension. / Materials Science and Engineering A362 (2003) 81–105 Fig. (3) stacked foam plates. 15. (2) foam plates compressed with different compaction ratios. With both methods a well controlled macroscopic porosity distribution inside the foam with minimum values of 40% open porosity and maximum values of 97% open porosity can be 91 Fig. 15 illustrates the processing steps on the example of a graded Cu/Al2 O3 material for burner rig testing [108].B. (6) preform material cut to specimen dimensions. 14). The volume content of the reinforcing fibers could be continuously varied between 25 and 60% Fig. Higher particle concentrations and lower winding speeds result in a distinctly more pronounced accumulation of spacer particles between the fiber filaments and consequently lower fiber volume fractions. A quantitative description of the porosity evolution was possible by an electrokinetic model described in Section 5. In a further priority program project. 14. (8) specimen for burner rig test after grinding and polishing. 17). Fig. fiber preforms with graded fiber spacing were produced by impregnating carbon fibers with slurry [109]. 16 shows an illustration of the used impregnation set-up. Manufacture of a Cu/Al2 O3 FGM by the GMFC process (processing stages are numbered consecutively): (1) uncompressed polymer foam. (4) foam stack infiltrated with Al2 O3 . (7) retained Cu/Al2 O3 FGM after infiltration of the preform with molten Cu. A very versatile method to produce porosity graded sacrificial preforms is also the inhomogeneous compaction of polymer foams as first described by Cichocki et al. After electrochemical gradation and infiltration with an alumina slip of sufficiently small particle size. Microstructure of a graded W/Cu composite produced by electrochemical gradation. 16. Fig. rate of dissolution and porosity of the anode. Electrochemical gradation was also used to produce porosity graded ceramic preforms. Porosity graded foams are produced by warm pressing foam wedges of different shape to a homogeneous thickness or by stacking compacted foam plates with different compaction ratios. In this case porous carbon was used as an anode material and sacrificial preform. In this so-called “gradient materials by foam compaction” (GMFC) process open-celled polymer foam of a low initial density is used as a starting material. produced [106. . After infiltration of the fiber preforms with different magnesium alloys carbon fiber reinforced composites with graded fiber content were obtained. After anodization the tungsten preforms were infiltrated with molten copper applying 10 MPa Ar pressure to yield a fully dense W/Cu FGM (Fig. Electrochemical gradation has been used to introduce porosity gradients in materials such as bronze [101] and copper [102]. Schematic diagram showing the impregnation process used for the introduction of spacer particles into the fiber bundle. the polymer foam is burnt out and the ceramic is sintered. Compared to ceramic or metal based systems. After centrifugation the filler content in the outer portions of the rings increased up to 27% for glass fibers and to 45% for SiC particles. If complete reactive infiltration is used to produce gradients. the composition profile of the product can be controlled by the gradient in the preform. This method was studied in detail by Klingshirn et al. and Al2 O3 and the respective metal aluminides are formed according to the reaction aMeu Ov + bAl → cAl2 O3 + dMex Aly (4) For the production of FGMs by the i-3A process graded Al2 O3 /TiO2 and Al2 O3 /Fe2 O3 bodies were produced by powder stacking or slip casting. compositional and microstructural gradients are intended to allow an optimum combination of component properties. impact resistance and toughness. In polymer composites. wear resistance. Processing of polymer-based FGMs In polymers. / Materials Science and Engineering A362 (2003) 81–105 Fig. Influence of the winding speed on the fiber volume content for several Al2 O3 particle concentrations. Mullite/alumina [110]. An example is the so-called i-3A process [113]. for an average filler content of 20%. The latter could be increased by gelation for a given time before . for example weight. whereas the amount of spacer particles in the fiber bundle decreases. The fiber content increases with higher winding speeds. by adjusting the winding speed during the impregnation process. the graded green bodies were converted into Al2 O3 /TiAl3 and Al2 O3 /FeAlx FGMs [114]. Polymers with a porosity gradient. 4. Graded polymers that have been processed so far include graded fiber composites [116]. This significant difference between different filler materials was primarily attributed to the aspect ratio of the filler particles. Rings and tubes with a gradient in filler content were produced by centrifuging a dispersion of particles in a resin prior to hardening. In the following. After gas-pressure infiltration with Al. some important processing methods for polymer FGMs are described where particular attention is given to the methods used in the priority program. aramide particles. Kieback et al. Partial infiltration is usually combined with a reaction and sintering step after infiltration in which a dense body containing the new phases is formed. The gradient may be formed by incomplete infiltration of a homogeneous porous body or by complete infiltration of a preform that contains a porosity or composition gradient.6. graded biodegradable polyesters [118] and graded index polymer fibers and microlenses [119]. A smooth gradient in the filler concentration could be produced by adjusting the rotation speed of the centrifuge and the viscosity of the resin. Reactive infiltration In reactive infiltration the preform reacts with the infiltrated material to form a graded composite. surface hardness.% and had defect-free interfaces between layers of different composition. Partial infiltration is a very simple method to produce continuous gradients. but lacks flexibility in the shape of the composition profile. a gradient of the reinforcement can be introduced by centrifuging prior to polymerization [120]. 17.92 B. 3. like in other materials. so-called polyurethane integral skin foams provide high impact strength at low weight and have been used since long for instrument panels or head rests in cars [115]. The advantages of reactive infiltration are low-cost processes and raw materials and improved wetting. The aluminide content in the FGMs varied between 0 and 35 vol. The metal oxide is reduced by the liquid aluminum. mullite/aluminumtitanate/alumina [111] and alumina/calcium-hexaluminate [112] FGMs were formed by partially infiltrating precursors such as silica sols or calcium acetate into alumina preforms. which was about 200 for the glass fibers and 1 for the SiC particles. glass fibers and carbon fibers were used as filler materials. in a priority program project [121]. the knowledge on processing methods for polymer FGMs is limited. in which Al is infiltrated into porous preforms consisting of Al2 O3 and a metal oxide. graded interpenetrating polymer networks [117]. The average filler content was chosen between 5 and 45%. while SiC particles. Two resins were used as matrix materials. Graded microstructure of sintered Vectran® fibres. the liquid-crystalline copolyester Vectran® . 5. Modeling of FGM processing The rapidly increasing performance of computers and computer software has favored the widespread use of numerical simulation tools for the design of parts and for the optimization of manufacturing processes. 18 shows an example of a graded microstructure with a nearly homogeneous structure at the high-temperature side and clearly discernible fibers at the low-temperature side. process modeling may help to circumvent or at least minimize problems such as warpage or cracking during the fabrication of a component. The gradient was achieved by hot press lamination of a stack of glass fiber prepregs containing differently modified resins. so that small objects can penetrate through 93 the material between the fibers and the high strength of the fibers cannot be used for the absorption of impact energy. In this test. This was done by dynamic mechanical analysis (DMA) which proved that a smooth gradient was established during molding [125]. Low sintering temperatures lead to an anisotropic microstructure. . it was necessary to determine the resulting gradient at the end of the process. Modeling of gradient formation In many processes such as powder stacking the gradient is formed directly and there is no need to model the formation Fig. but only when the anisotropic microstructure was at the impact side and the isotropic structure is at the opposite side. and cross-ply rather than fabric was used [126]. the idea was successful. For one of the test materials. Long-fiber reinforced polymers with graded fiber composition or orientation can be produced by lamination techniques [124]. / Materials Science and Engineering A362 (2003) 81–105 centrifugation. The wear resistance of thermosetting polymers was improved by the addition of the hard filler particles or fibers. in which the aligned fibers are still well discernible. Fig. In some cases the simulation can help to establish the desired gradient. Sintered fiber materials were selected as candidates because they have high energy absorption potential.123]. it was modified by a rubber modifier.1. Polycyanurate resins were used as a matrix material. Components with either of these microstructures have poor to moderate impact resistance. In a priority program project a gradient was introduced also in the matrix material. Since the neat polycyanurate is too brittle for many structural applications. were obtained with the graded material. Such an effect was used in a priority program project aiming at the development of a polymeric impact protection material consisting of a single component. per se. Higher modifier contents improve the toughness. Process modeling may be especially important in connection with functionally graded materials. The idea of the project was to introduce a microstructural gradient by applying a temperature gradient during sintering with the goal to improve the impact properties of the component. If polymers develop distinctly different microstructures at different processing temperatures. since gradients often cause special problems. The idea of the project was thus to optimize the properties of a component by a gradient in modifier content. The impact resistance was measured in a repetitive drop-weight test. Since the resins are redistributed between the prepreg layers during molding. 5. In other cases. in addition to gradients in fiber density and orientation which are already used in lightweight components. it is possible to introduce property gradients by applying a temperature gradient during processing. isotropic material. On the other hand. in which the plates are impacted repeatedly with increasing impact energy until penetration occurs. the homogeneous isotropic material exhibited very low energy absorption values around 20 J. Kieback et al. but the good high-temperature stability of neat polycyanurate and the resistance against water absorption are reduced. A viscosity 25 Pa s. while the homogeneous material with anisotropic structure (sintering temperature 290 ◦ C) gave energy absorption values around 200 J with a small difference between cross-ply and fabric. It was demonstrated that the gradient in filler content leads to a corresponding gradient in microhardness and wear resistance [122. nearly 300 J. and to maintain it throughout the entire process by an appropriate process control. The best values. was found suitable for centrifugation of carbon fibers—a lower viscosity of 5 Pa s lead to a complete sedimentation of the carbon fibers during thermosetting. The properties of the final product depend strongly on the temperature at which the fibers are sintered together.B. the isotropic microstructure. has low impact resistance. 18. High sintering temperatures lead to a compact. The anisotropic material fractures by delamination between the fibers. Further. A finite-difference scheme based on the same differential equations as the analytic solution was developed in order to be able to use arbitrary current–potential relations. For a plate-like porous anode of thickness L where the current feeder is located at the side opposite to the cathode (Fig. 19 shows [132]. The results of the analytical and the numerical model were compared to experimental data for a porous tungsten electrode (Fig. The produced porosity distribution can be controlled by a number of experimental parameters (Fig. For such processes the relationship between composition of the gas phase and the composition of the deposit may be complicated and knowledge of the relationship between the partial pressures of different components in the gas phase and the film growth rate and composition is required for the design of controlled gradients [131]. Normally. as Fig.130] and reasonable agreement with the experimental particle distribution was found [130].128]. which means lower density. but the numerical model could predict the porosity distribution even with a higher precision. Kieback et al. while the core of the part has low density. This is achieved by keeping the tool surfaces relatively cool (typically 80 ◦ C). Already the analytical model showed a good agreement with experimental data. When the powders are not deposited directly but in a filtration process knowledge of the rate of cake formation is necessary to produce pre-defined gradients. It was assumed that the local density of the Faraday current per unit internal surface area jF is governed by the Tafel equation: jF (η) = j0 exp αF η RT (5) where j0 and α are constants for a given electrode reaction and η the overpotential (driving the electrode reaction). Increasing the total current and keeping the other experimental parameters constant lead to a more uneven tungsten distribution (compare a and b). / Materials Science and Engineering A362 (2003) 81–105 of the gradient. Eq. the Stokes equation is used to describe the particle velocity. In the following a model is presented that allows an accurate prediction of the porosity gradient developing during the electrolysis due to the dissolution of the electrode material. Polyurethane integral skin foam parts exhibit a substantial density gradient. Porosity distributions produced by the electrochemical gradation process were thus predicted quantitatively in a numerical model of the gradation process taking into account .94 B. Predicted and optically measured density distributions were consistent. a priority program project using anodic dissolution to prepare graded porous preforms for infiltration processing was described. As material is removed by the anodic reaction a position-dependent porosity and specific surface area develops. 21). In vapor deposition processes the gradient is usually introduced by dosage of precursors from separate containers or by adjusting the partial pressure of a reactive gas. Preforms with lower initial porosity graded under the same experimental conditions also showed more uneven tungsten content (compare b and d). whereas the center of the part is heated by the foaming reaction to higher temperatures (typically 180 ◦ C). In a priority program project the foaming process was simulated numerically using the finite-element code ABAQUS. (6) is only valid if the current potential relation is described by the Tafel relation. but for heavier particles Newton’s law has been used [58]. It has been demonstrated by Tuffé and Marple that a precise control of gradient formation is possible using a simple analysis if the casting parameter is known for each slip composition [127]. modeling of particle sedimentation is straightforward if the particle size distributions are known. In the previous chapter. A continuum model was used which regards the porous electrode as a macrohomogenous mixture of the electrode matrix and the electrolyte filling the pores. and how the element could be fabricated. a new method was developed to detect the gelation of the foam material by ultrasound transmission and a suggestion was made how a graded foam structure could serve as a coupling element for ultrasound between air and a sensor. such that the surface is relatively dense and mechanically stable. A and B are dimensionless constants defined by the following equations: αF A = (ρl∗ + ρs∗ )L2 Sj0 (7) RT √ B C D (8) arctg √ + arctg √ = 2 B B with αF αF and D = ρl∗ jT L (9) RT RT The position of minimum overpotential is determined by the dimensionless quantity wmin : C 2 wmin = √ arctg √ (10) B B C = ρs∗ jT L ρs∗ and ρl∗ are the effective resistivity of the porous electrode and electrolyte. Reasonable agreement of calculated and observed particle volume fractions have been achieved for sedimentation experiments [59. which depend on the pore structure of the electrode and the resistivity of the employed materials. Increasing the resistivity of the electrolyte by decreasing the concentration of electrolyte also leads to a more uneven tungsten distribution (compare c and d). 20). This leads to an expansion of the foam bubbles to much larger diameters. As long as particle interactions are small. 13) the current density per unit volume jv (and thus the local rate of dissolution) could be determined analytically as: √ B B jv = SjF = j0 S 1 + tg2 [w − wmin ] (6) 2A 2 where the relative position w = x/L and the surface area per unit volume S have been introduced. Modeling of gradient formation has also been carried out for centrifugal casting [129. This was also included in the numerical model by the dividing the operation time into small time intervals and adding of the material removal during all time intervals. / Materials Science and Engineering A362 (2003) 81–105 average density 900 800 density [kg/m³] 294 kg/m³ 376 kg/m³ 480 kg/m³ 597 kg/m³ 680 kg/m³ 95 700 600 500 400 experiment simulation 300 0 4 8 2 6 distance from upper surface [mm] 10 Fig. 22 and 23 demonstrate that the evolution of the strains (and hence of the density) and of the grain size can be described well by the model.B. and the axial and radial strains are measured. Figs. 5. Kieback et al. and they were also combined with beam-theory and plate-theory PC programs in order to predict the warpage and cracking of graded components. a cylindrical specimen is subjected to an axial load during sintering. Table 1 shows the resulting parameter values. since for relatively low liquid contents the structure of the liquid-phase model is similar to that of the solid-state model. the . 5. This is not surprising. It should be noted that a similarly good fit of the data can be achieved by a solid-state sintering model [133]. 19. A linear heating rate of 5 K/min and a hold temperature of 2173 K are prescribed with a linear temperature gradient of 100 K over the plate thickness during the rise time. until the simulated behavior agrees with the measured one. Application of a liquid-phase sintering model to Al2 O3 In [134. which was used in this project to describe the sintering behavior of 94% purity Al2 O3 . To determine the parameters of the model.2. (.%. Second. Finally a program for microwave sintering was developed and combined Fig. models for solid-state and liquid-phase sintering were developed and applied in the priority program [133–135]. In this test. Modeling of sintering As a basis for describing the sintering of graded components. (—) numerical model. The numerical model includes the effects of structural changes on the current distribution in the porous electrode. since the models contain the grain size as an internal state variable. Measured and calculated density distributions across the thickness of a polyurethane plate after the foaming reaction. microstructural changes of the porous electrode during the dissolution process. Initial tungsten content was 75 vol. The equations for the equilibrium of force and bending moment were given in [136].2 mA/cm2 and NaOH concentration 2 mol/l.2. First a relatively simple program was written in the framework of plate-theory. As an example. The lower half of the plate consists of molybdenum and the upper half of ZrO2 . Warpage of graded plates The warpage of graded plates during sintering was modeled in two ways. current density 12. the sinter warpage of a plate with a grain size gradient was calculated with a finite-element code [62.2. 20.2. The models were implemented in a finite-element code (ABAQUS). In addition the evolution of the microstructure was observed. The models and their application in the priority program are described in the following paragraphs. with an advanced solid-state sintering model. 5.135] a liquid-phase sintering model was developed.1. a circular plate with 10 mm thickness and 100 mm diameter is considered in [136].136].-) analytical model. Comparison of the predicted and observed tungsten content in an electrochemically graded W/Cu composite. The parameter fit is done using a PC program by integrating the evolution equations of the model and varying the parameters by hand.. so-called sinter forging tests were carried out. (d) initial porosity 42%.25 J/m2 0. (c) initial porosity 42%. current density 5. Kieback et al.1 Grain rearrangement Maximum solid density by rearrangement Viscosity. activation energy 5.25 J/m2 0. cNaOH 2 mol/l.5 mA/cm2 .5 mA/cm2 .63 1 × 105 425 kJ/mol Solution/re-precipitation Reaction rate.12 1 Ratio shear-to-bulk viscosity G/K 0.5 mA/cm2 .27 .2 mA/cm2 . activation energy 0. current density 5. current density 12. (b) initial porosity 25%. cNaOH 2 mol/l.2 × 1038 425 kJ/mol Sintering stress Solid/liquid interface energy Liquid surface energy Volume fraction of large pores Parameter for size distribution of large pores B Parameter for size distribution of large pores n 0. / Materials Science and Engineering A362 (2003) 81–105 TUNGSTEN CONTENT [Vol.1 0.-%] 3 4 2 (b) 6 TUNGSTEN CONTENT [Vol.-%] 96 (d) Fig. Open triangles are hardness values of the composites: (a) initial porosity 25%. Comparison of the predicted (—) and observed (䉬) tungsten distribution in electrochemically graded W/Cu for different current density. pre-exponential factor Viscosity. pre-exponential factor Diffusion coefficient. NaOH concentration and initial porosity. activation energy 3.60 2 40 1 20 2 4 0 80 3 60 2 40 1 20 0 2 60 2 40 1 20 0 0 2 POSITION [mm] (a) TUNGSTEN CONTENT [Vol. 21.05 m Melt formation Start of melting Melting rate Maximum volume fraction of melt 1430 K 0.605 1.-%] 6 4 6 0 VICKER'S HARDNESS [GPa] 0 80 3 4 6 (c) 3 80 2 60 40 1 20 0 0 TUNGSTEN CONTENT [Vol. pre-exponential factor Reaction rate.9 × 104 400 kJ/mol Diffusion in liquid phase Diffusion coefficient.-%] 80 VICKER'S HARDNESS [GPa] B. cNaOH 2 mol/l. cNaOH 1 mol/l. current density 5.000595 K 0. Table 1 Parameters of the liquid-phase sintering model for 94% purity alumina Initial values Relative green density D0 Initial grain radius R0 0. but slightly later also the ZrO2 shrinks and just compensates the initial warpage. Kieback et al. It was found that up to 800 ◦ C.5 2. and 3 m at the coarse-grained side (as measured by quantitative image analysis). the incompatibilities between areas sintering at different rates . The initial grain radius is chosen as 2. / Materials Science and Engineering A362 (2003) 81–105 40 -0. These predictions are consistent with observations of Moritz et al. cracking is a potential danger.1 radial strain -0. Hence the plate is substantially distorted at the end. 23. 310 kJ/mol for ZrO2 . 25).2. A three-dimensional finite-element calculation. depending on the material properties and the geometrical conditions. Qm .74 in ZrO2 and 0. Warpage of a plate consisting of molybdenum and zirconia. Solid lines: experiments. Temperature Mo-ZrO2 ∆T = 100 K 400 Time in min 2. if the zirconia has a grain radius of 1 m. Softening may lead to strain localization.6 for Mo and 0. Evolution of axial and radial strains. Within the framework of plate-theory the distortion necessarily remains axisymmetric. Symbols: experiments. since different regions sinter at different rates. With increasing sintering temperature the plates were distorted.1 µm 10 R= 0. Qb . Fig. 23. It is suggested that cracking is initiated in a sintering body if the mean stress locally exceeds the sintering stress. The resulting incompatibilities may either be relieved by creep accommodation and distortions. The deformation is only axisymmetric for small deflections. whereas the plate assumes the form of a potato chip as sintering progresses [136].2 free sintering Displacement in mm Strain 0. and the pores were still open with pore sizes between 5 and 50 nm at the fine-grained side. dashed lines: model. Later also the Mo starts to sinter. shows that the initial symmetry is spontaneously broken. The final warpage is small.5 m for Mo. however. The sinter warpage could be avoided without suppressing the radial shrinkage by covering the graded plate with an alumina disc of about 3 g mass. since intense grain coarsening prevents further densification. but it does not nearly reach full density. or they may lead to cracking.B.0 1400 Experiment 1300 1200 Model 1100 Temperature in ˚C Grain Size in µm 3.3 35 N axial strain -0. 0. In this case the material starts to desinter. They are taken from other investigations to be Qb = 280 kJ/mol for Mo. Prediction of cracking When graded materials are processed. At about 850 ◦ C the plates developed a certain mechanical strength. 24 shows the calculated evolution of the warpage of the plate (the normal displacement of the edge of the plate with the center being fixed).88 in Mo.5 and 1 m). 24.5 µm R= 1 µm 0 -10 0 Fig.1. This leads to a displacement of the plate edge in upward direction.5 200 250 300 40 N 350 ZrO2 side being cooler. Temperature is the same as in Fig. The relative green densities are assumed to be 0.3. practically no sintering occurred. This distortion can be minimized by choosing the appropriate grain size for the ZrO2 powder. If the grain radius of the ZrO2 powder is 1 m the Mo side starts to sinter first. 2 4 6 8 10 12 14 Time in h Fig. but still have an open pore structure. 22.0 200 97 400 Time in s Fig.4 45 N -0. In this case the final relative densities are 0.5 for ZrO2 .5 1000 250 300 350 30 20 Grain radius of ZrO2 R= 0. 5. and that of ZrO2 is varied (0. when sintering pore-size graded titania plates [62]. dashed line: model. Evolution of the grain size. 271 kJ/mol for ZrO2 and Qm = 230 kJ/mol for Mo. even in the fine-grained top layers.0 -0. The sintering temperature was selected such that the plates acquire enough strength. Part of the warpage was reversed when also the coarse-grained material sintered in the temperature range between 900 and 1200 ◦ C. The fine-grained ZrO2 sinters at a temperature where Mo is still practically inactive. The material parameters that are most important for the sintering behavior of the plate are the activation energies for grain boundary diffusion. 1700 Al2O3 A1894 1600 1500 3. as the fine-grained side sintered faster than the coarse-grained side (Fig. which means that it locally softens. and for grain boundary migration. a transition layer with intermediate properties between the core and the bulk is less favorable for the sinterability than a sudden transition. which is intended Fig. 26 shows the result of the calculation for three different grain sizes in the interface layer (cases a. Fig. (b) 0. the configuration shown in Fig.52 and 1. In further calculations. Tensile stresses develop between the coarse-grained bulk and the fine-grained core due to the different sintering stresses.52 m.522 m.5 and 2 m. the principle can be generalized to arbitrary geometrical configurations with property gradients. to serve as a perturbation to initiate cracking. if the grain size in the interface is close to either that of the core (0. The question to be answered is whether the greater sinter activity of the fine-grained core leads to a decohesion from the bulk or whether the core remains attached. desintering leads to a complete loss of density and strength of the interface layer. In case (a) the grain size in the interface layer is 0. 26 is considered: A spherical core of fine-grained material (region 1) is separated by a thin layer (region 2) from the coarse-grained bulk (region 3). as well as the complete system densify readily.518 m. . 26. The initial relative density is chosen as 0. When the interface layer breaks down.518 m. In the framework of the finite-element method.52 m. (B) Evolution of the relative density in the fine-grained core.54. the core starts to sinter much faster than before when it was still constrained by the slowly sintering. b and c). in the interface layer and in the coarse-grained bulk. In case (c). a seemingly minor change of the grain size in the interface leads to a completely different result. Kieback et al. / Materials Science and Engineering A362 (2003) 81–105 Fig. In case (b) the grain size in the interface layer is 0. where it is 0. the interface.522 m. To be specific the grain size in the bulk is assumed to be 2 m. decohesion occurs. If the interface has grain sizes between 0. in the core it is 0. coarse-grained bulk. in the evolving crack. It turned out that stable sintering is only possible. (c) 0. its sintering rate being constrained by the slowly sintering bulk. however. the grain size of the interface layer was varied.5 m and in the interface layer it is given a value between 0. Grain size in the interface: (a) 0. This example is intended to illustrate the principle how cracking can be described as a process of localized desintering for a simple configuration.98 B.e. Influence of the sintering temperature on the warpage of graded titania plates.95 m. (A) Configuration analyzed in the sintering model. where d = 0. near the instability point. are accommodated in a narrow desintering layer. Still the interface remains intact and reaches full density after a period of initial desintering. Hence in this case. To illustrate the principle. while all other parameters remained fixed. After a period of retarded densification. Hence.55 except in the interface layer. i.5 m) or that of the bulk (2 m). The numerical values for the grain sizes are deliberately chosen to illustrate the sudden transition from stable sintering of the system to unstable desintering (cracking) of the interface. These stresses act on the interface layer. 25. Cylindrical microwave cavity with a ceramic sample and susceptors. the electromagnetic field in the microwave cavity is inhomogeneous. 27 shows the axisymmetric microwave furnace to be analyzed. The susceptors are assumed to be dense alumina. and the program developed here takes that also into account [137]. Commercial microwave codes generally solve Maxwell’s equations. 28. In the case shown here the temperature reaches the melting point of alumina in the hot spot. The microwave frequency is chosen such that the TE031 mode is excited (for the microwave terminology see [138]). As Fig. 29 shows. but it may be critical for the success of a sintering process.B. Apparently the conditions for a run- Susceptor Ceramic Insulation Microwaves Antenna Fig. if fine-grained alumina with grain size 0. and the insulation is given the thermal and electric properties of alumina fibers. / Materials Science and Engineering A362 (2003) 81–105 5. 27.1 m is sintered with 500 W microwave power. In addition to the temperature dependence. This leads to a density near 100% and a fine grain size below 0. On the other hand. Since the heat conductivity increases sharply during the initial stages of densification. 28 shows the temperature distribution for the case that a coarse-grained alumina with grain size 2 m is heated with 1 kW microwave power.2. Such a runaway instability can occur if the microwave absorption coefficient increases with temperature. which is an additional difficulty for the sintering of graded parts. 99 Fig. In practice this means that the process fails to yield an acceptable sintered product. away instability to occur are satisfied here. if the heat conductivity is low. The following example shows that the full coupling of the sintering model with the electromagnetic and thermal fields is not a second-order effect. The reason for the favorable behavior of the fine-grained material is that it starts to sinter earlier. (b) Maximum and average temperature in the (coarse-grained) ceramic show the evolution of a runaway instability. The material to be sintered is an alumina ceramic. the heat is spread more effectively and hence no localization instability develops.4.14 m throughout the specimen. the temperature distribution in the fine-grained ceramic is very homogeneous.and fine-grained . A model which does not couple sintering with the electromagnetic and thermal fields would not be able to predict the difference between coarse. sometimes coupled with the heat conduction equation. (a) Temperature distribution for 1 kW microwave power. while the antenna at the top serves to control the process. On the other hand. where it quickly rises to very high values. The picture shows that a so-called runaway instability occurs: the temperature is rather low everywhere in the ceramic (the hatched area in the center) except in a small region. Fig. if the microwave power is high. and if the specimen is large. the instability can be avoided. Kieback et al. Fig. for example for the sintering of fine-grained ceramic powders. when the fine grain size is to be preserved during sintering. while the rest of the ceramic does not even start to sinter. The energy is fed into the cavity by the bottom antenna. since the dielectric properties of materials usually depend on temperature. Microwave sintering Microwave sintering offers certain advantages compared to conventional heating. Hence a finite-difference microwave program was written and combined with the solid-state sintering model described in [133]. the thermal and electric properties may depend strongly on the sinter density. Nevertheless. latent heat) are taken into account. which causes the well-known shrinkage during drying. Functionally graded materials are especially susceptible to warpage during drying. On the other hand. materials. It turned out that under all practical drying conditions the temperature was nearly uniform for a wall thickness of a few centimeters. namely the local water content. accessible to several finite-element codes.3. the latent heat of the evaporating liquid and the internal energy in the body. The thermodynamic properties of water (vapor pressure over curved surfaces. The development of stable microwave sintering processes for functionally graded materials without local over-heating can be substantially facilitated by the use of simulation tools like the present one. The capillary pressure of the liquid exerts an increasing pressure on the solid skeleton as the liquid content decreases. which leads to distortions and. Various aspects of drying theory are described in [139. Since the rate of drying and the capillary pressure depend on the grain size and the green density. since now only two fields (water content and deformation field) remain which must be treated in a coupled manner. Water content and shrinkage during drying. 5. ponents with a commercial finite-element program.140]. in principle. Hence it is not necessary to solve the heat conduction equation. Measured values and adjusted theoretical curves. Modeling of drying processes Water content in wt-% 3 20 2 15 10 1 5 0 0 100 200 300 400 500 600 Shrinkage in % After wet forming processes it is necessary to dry the green powder compact. 29. namely the drying simulation itself and the deformation problem. A complete description of drying and the concomitant deformations involves three coupled fields. also to crack formation. The drying problem is then analogous to a heat conduction problem. the assumption is usually justified for industrial drying processes. This excludes the description of situations in which the water is pressed out of the porous solid by applied forces like in pressure filtration. and the deformation problem is subsequently solved as an elastic–plastic problem with superimposed volumetric strains and plastic properties that depend on the local water content. This two-field problem is. In a first step. It suffices to consider the global energy balance between the heat flowing into the body through its surface.100 B. a one-dimensional PC program was developed. Kieback et al. This greatly simplifies the analysis. (b) Maximum and average temperature in the (fine-grained) ceramic. In the priority program a basis for modeling the warpage during drying was developed. the shrinkage is usually inhomogeneous in a graded component. (a) Temperature distribution for 500 W of microwave power. The program simultaneously solves the heat conduction equation and Darcy’s equations for liquid and vapor transport in the porous solid. / Materials Science and Engineering A362 (2003) 81–105 Fig. before it can be processed further. in some cases. . the problem was further decomposed into two one-field problems. The decoupling is based on the assumption that the hydrostatic part of the stress is small compared to the capillary stress of the liquid. 30. Hence in a numerical time-stepping algorithm the (uniform) temperature change can be calculated in each step directly from the current state of the body and from the temperature and humidity of the surrounding atmosphere. The aim was to be able to model the drying and deformation behavior of graded com- 25 0 700 Time in min Fig. the temperature and the mechanical deformation. Maximum thickness is limited. L C C M T C C C C C M. Initial geometry in the background. Mb UT. and the volumetric drying strain is assumed to increase linearly from 4 to 8% from the top of the plate to the bottom. The numbers give the range in which the water content varies. C Very good Very good Very good Very good Very good Very good Very good Very good Very good Very good Moderate Good Good Very good Moderate Bulk Bulk Bulkc Coating Bulk Bulk Bulkc Bulk. As an example the drying of Fig. CVD GMFC process Very good Very good Very good Very good Very good Good Very good Very good Very good Moderate Moderate Moderate Moderate Very good Very good M. Tb UT. C: continuous. Warpage after drying. Fig. Tb M. The material is given model parameters similar to those of a sanitary ware material. Distribution of the water content in the plate after 650 s (a) and 6600 s (b). Kieback et al. Table 2 Overview of processing methods for FGMs Process Variability of transition function Layer thicknessa Versatility in phase content Type of FGM Versatility in component geometry Powder stacking Sheet lamination Wet powder spraying Slurry dipping Jet solidification Sedimentation/centrifuging Filtration/slip casting Laser cladding Thermal spraying Diffusion Directed solidification Electrochemical gradation Foaming of polymers PVD. The parameters of the model were determined from a set of experiments. L. while the permeability for liquid water increases by a factor 3. Further. Evolution of the temperature in the plate during drying. coating Bulk Bulkc Bulk Coating Bulk Moderate Moderate Moderate Good Very good Poor Good Very good Good Good Poor Good Good Moderate Good a b c L: large (>1 mm). 32. After the determination of the model parameters the drying and warpage of homogeneous and functionally graded materials can be described. Depending on available powder size. a graded circular plate with 60 mm diameter and 5 mm thickness was modeled in an atmosphere with 25 ◦ C and 50% humidity. bulk Joint. stress–strain curves were measured in compression tests for different water contents to determine the elastic–plastic properties of the wet powder body. M: medium (100–1000 m). 31. / Materials Science and Engineering A362 (2003) 81–105 101 30 28 temperature in ºC 26 24 22 20 18 16 14 12 10 0 5000 10000 15000 20000 25000 30000 35000 40000 time in s Fig.B. L T. 30). coating Coating. UT: very thin (<10 m). Only a quarter of the plate is depicted. 33. . in which the water content. the shrinkage and the temperature are measured for different temperatures and humidities of the atmosphere (Fig. T: thin (10–100 m). C. Sci. Germany.-P. H. 17B. I. Sendai. pp. p. Z. Ilschner. in: Proceedings of the Euro PM99. Presses Polytechniques et Universitaires Romandes. vol.6%. Italy.-C. Toda. M. but also on the type and extension of the gradient. Mater. Koizumi. Met. Kieback et al. R. Bever. Dreyer. 1995. 32 shows the distribution of the water in the plate at different times. J.). Lengauer. UK.000 s the water content in the plate reaches 1. W. M. Functionally Graded Materials. Mortensen. M. Tokita. Mater. Final Report. H.J. in: M. Powder Metallurgy—Processing and Materials. 1990. Y. J. in: Proceed- . Cherradi (Eds. Proceedings of the 3rd International Symposium on Structural and Functional Gradient Materials. Chen. von Ruthendorf. Kluwer Academic Publishers. 16 (1987) 9–15. 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Oh. in: Proceedings of the 14th International Plansee Seminar. 463–473. R. it is in equilibrium with the atmosphere and evaporation comes to a standstill. Hirai. Yamanouchi. Ilschner. VCH Verlagsgesellschaft. since the capillary stresses of the liquid in fine-grained material are much larger than in coarse-grained material [141]. 275. K.F.).-S. planar plates could be obtained by supercritical drying. Proceedings of the 5th International Symposium on FGM. Refract.). Düsseldorf. Kawahara. In practice. T. 83–88. M. Sendai. Switzerland. Sintervorgänge. Switzerland. B. Chen. Res. 32 (1997) 3841–3850. pp. Proceedings of the 1st International Symposium on Functionally Gradient Materials. 101–106. J. Schatt. especially in freeform processing and metallurgical processing of FGMs. Kaysser (Ed. Brook (Ed. N. (iii) cost-effectiveness of production processes. Green. Koizumi. Shen. Ford. B. B. T. [26] [27] M. Dresden. 1990. 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