processing techniques for functionally graded materials9... · materials science and engineering...

Materials Science and Engineering A362 (2003) 81–105

Processing techniques for functionally graded materials�

B. Kiebacka, A. Neubrandb,c,∗, H. Riedelc

a Institut für Werkstoffwissenschaft der TU Dresden, Mommsenstraße 13, 01062 Dresden, Germanyb 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 processingtechniques is given. Established powder processes and techniques involving metal melts are described, and recent developments in the fieldof graded polymer processing are considered. The importance of modeling of gradient formation, sintering and drying for the production ofdefect-free parts with predictable gradients in microstructure is discussed, and examples of a successful application of numerical simulationsto 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 propertieschange gradually with position. The property gradient inthe material is caused by a position-dependent chemicalcomposition, microstructure or atomic order. In the caseof a position-dependent chemical composition the gradi-ent can be defined by the so-called transition functionci(x,y, z) which describes the concentration of the componentci as a function of position. Already in 1972, the useful-ness of functionally graded composites with a graded struc-ture was recognized in theoretical papers by Bever andDuwez [1], and Shen and Bever[2]. However, their workhad only limited impact, probably due to a lack of suit-able production methods for FGMs at that time. It took15 more years until systematic research on manufactur-ing processes for functionally graded materials was carriedout in the framework of a national research program onFGMs in Japan. Since then, a major part of the research onFGMs was dedicated to processing of these materials and

� This is an overview of parts of the key results obtained within thePriority Program “Functionally Graded Materials” funded by the GermanResearch Foundation (DFG) in the years 1995–2002.

∗ Corresponding author. Tel.:+49-761-5142282;fax: +49-761-5142403.

E-mail address:[email protected] (A. Neubrand).

a large variety of production methods has been developed[3–6].

The manufacturing process of a FGM can usually bedivided in building the spatially inhomogeneous structure(“gradation”) and transformation of this structure into abulk material (“consolidation”). Gradation processes can beclassified into constitutive, homogenizing and segregatingprocesses. Constitutive processes are based on a stepwisebuild-up of the graded structure from precursor materialsor powders. Advances in automation technology during thelast decades have rendered constitutive gradation processestechnologically and economically viable. In homogenizingprocesses a sharp interface between two materials is con-verted into a gradient by material transport. Segregatingprocesses start with a macroscopically homogeneous mate-rial which is converted into a graded material by materialtransport caused by an external field (for example a gravita-tional or electric field). Homogenizing and segregating pro-cesses produce continuous gradients, but have limitationsconcerning the types of gradients which can be produced.

Usually drying and sintering or solidification follow thegradation step. These consolidation processes need to beadapted to FGMs: processing conditions should be chosen insuch a way that the gradient is not destroyed or altered in anuncontrolled fashion. Attention also has to be paid to unevenshrinkage of FGMs during free sintering. Since the sinteringbehavior is influenced by porosity, particle size and shape

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82 B. Kieback et al. / Materials Science and Engineering A362 (2003) 81–105

and composition of the powder mixture, these problems mustbe handled for each materials combination and type of gradi-ent individually referring to the existing knowledge about thesintering mechanisms[7]. It has been demonstrated that theintroduction of a controlled grain size gradient in additionto the composition gradient can balance different sinteringrates[8]. In general, this optimization results in unjustifiablelimitations for the gradients design. Additionally, variationsin the particle packing density can lead to deformation of thepart. To overcome some of these limits, the superpositionof the internal driving forces for sintering with an externalpressure by hot pressing[9–11] or hot isostatic pressing[12] is possible. If the temperature regions for densificationare very different, application of a temperature gradientduring consolidation[13], liquid-phase sintering[14], laserassisted sintering[15], and spark plasma sintering[16,17]have been proposed. Although many of these methods forfunctionally graded materials were already developed in theearly 1990s, there were still limitations concerning materialcombinations, specimen geometry and cost in 1995. Oneof the goals of the German priority program “FunctionallyGraded Materials” was thus the improvement of existing andthe development of new processing techniques for function-ally graded materials. Besides the well established powdermetallurgical techniques other production processes suitablefor metals and polymers with a low melting point were in-vestigated in the priority program. Particular emphasis wasalso placed on modeling of production processes. Processsimulations may allow the prediction of suitable processingparameters for FGMs in the future and reduce the consider-able amount of experimental effort which is still necessaryto produce a graded material free of macrodefects. In thefollowing sections on powder metallurgy, melt and polymerprocessing and modeling, the achievements of the priorityprogram are described. Each field is introduced with basicbackground information and literature, but a comprehen-sive literature review of the international literature is notattempted.

2. Powder metallurgy

The powder metallurgy (PM) or ceramic technology routefor processing of materials and engineering parts includespowder production, powder processing, forming operationsand sintering or pressure assisted hot consolidation. Pow-ders of many metals, alloys, compounds and ceramic mate-rials with particle sizes ranging from nanometers to severalhundred micrometers are available from industrial sourcesor may be produced by the methods developed over decadesin the field of PM[18] or ceramics. Since the material usedto form the sintered part is dispersed into very small por-tions of individual powder particles nearly ideal conditionsexist to build-up graded materials with varying chemistryor microstructure by using the particles as building blocks.Practical considerations towards an efficient process design

suggest the use of powder mixtures with changing averageparticle size or composition during deposition of the mate-rial prior to the forming operation. It depends on the methodapplied for powder deposition whether a smooth change ora stepwise variation is obtained in the green body. Sincethe consolidation of the green parts during sintering or hotpressing requires high temperatures at which diffusion pro-cesses are fast enough to enable densification, chemical re-actions between particles of different compositions and theinfluence of particle size on the sintering behavior[7] haveto be taken into account for the final microstructure and di-mensional control of the part. On the other hand thermody-namic factors acting during sintering may be used to createa gradient, e.g. during liquid-phase sintering.

By the powder metallurgy route the following types ofgradients can be processed:

(i) Porosity and pore-size gradients: Porosity gradientsmay be created either by the deposition of powder mix-tures with different particle shape or by varying thedeposition parameters including the use of space hold-ers. Pore-size gradients in general are produced by thevariation of particle size. The different sintering behav-ior of the powders requires special attention to managethe problems of part distortion but may also be used toproduce a porosity gradient.

(ii) Gradients in chemical composition of single phase ma-terials: The deposition of powders with a continuouschange in the composition of the mixture during sinter-ing will lead to a single phase material with a smoothchange in the distribution of elements if the phase dia-gram shows solubility over the chosen range of compo-sition. The condition that has to be fulfilled is that thediffusion length of the different atoms is bigger thanthe average particle size or the characteristic width ofthe layered structure if stepwise deposition is applied.The starting powders may be elemental or chosen inthe range of compositions between which the gradientshould be formed. No application has been reported fora single phase material so far, but gradients in the mi-crostructure and properties after cooling and heat treat-ment caused by small changes in the elemental compo-sition might be interesting for intermetallic phase mate-rials based on gamma titanium aluminide. The amountof �2 and�-phase in these two phase materials and thetype of microstructure (lamellar, globular or duplex)obtained after thermal treatment depend directly on theTi/Al ratio.

(iii) Gradients of the volume content of phases and grainsize gradients in two or multiphase materials: Mostof the published literature focuses on gradients of thevolume content of different phases. The phase diagramof such a multi-component system is characterizedby no or limited solubility. In the latter case graingrowth of the soluble phase during the heat treatmenthas to be taken into account, in particular if a liquid

B. Kieback et al. / Materials Science and Engineering A362 (2003) 81–105 83

phase is used for activated sintering. The resulting mi-crostructure consists of two or more phases with gra-dients in volume content or in grain size which werepredetermined during the deposition of the powders.The combinations of phases can include metal–metal,metal–ceramic and ceramic–ceramic systems. Mi-crostructural gradients can also be produced by con-trolling the phase equilibrium during liquid-phase sin-tering with gradients of the components[19–24]. Thetechniques to obtain a graded material by the powdermetallurgy route[4,25–29]are classified in the follow-ing section according to their capability to produce astepwise or continuous change of the composition ofthe powder compact.

2.1. Formation of graded powder compacts

2.1.1. Deposition of layers of powder mixtures withstepwise changes in the mixture

2.1.1.1. Die compaction of layers (powder stacking).Inthis simple and well established method a gradient is formedby the deposition of powder layers with changing composi-tions in the compacting die[30–35]. The disadvantages ofthe process are obvious: discrete changes, limited numberof layers (up to 10 in laboratory scale, but not more thantwo or three in potential fabrication), limited thickness ofindividual layers (normally not less than 1 mm), limited sizeof the part (<100 cm2) due the limits of compaction forces,discontinuous manufacturing with low productivity. Never-theless this method allows effective laboratory studies offunctionally graded systems.

2.1.1.2. Continuous dry deposition of layers.A continu-ous process for the deposition of the powder mixture withchanging compositions on a conveyor belt was developed in[36]. Sequential deposition of powders results in a steppedgradient. Therefore a special synchronized distributor wasdesigned which makes a continuous change of the compo-sition feasible.

2.1.1.3. Sheet lamination.Thin sheets with different com-positions can be produced by dry or wet powder techniques(powder rolling[18] or tape casting) and joined to form astepped gradient[37,38]. Powder rolling is a well developedprocess giving green sheets with a thickness in the rangeof 1 mm. Tape casting of very fine powders even allows asheet thickness in the double-digit micrometer range. Thehigh productivity of the green sheet production offers a highpotential for up-scaling by laminating sheets with differ-ent compositions. The number of sheets would be limitedmainly by the costs of fabrication. Complex FGM parts canbe varied by stamping or laser-cutting the sheets to differentsizes[39]. For powder rolling only powders which result ingood green strength are suitable, while tape casting needs abinder that has to be removed before sintering. Hot press-

ing is used to join the layers during the final consolidationstep. In some studies[39,40], this step was accompanied bya simultaneous SHS-reaction.

2.1.1.4. Wet powder spraying.Powder suspensions suit-able for deposition with an air brush system were originallydeveloped to deposit thin powder layers on a substrate[41].By including a mixing system and controlled feeding of twoor more suspensions graded powder layers can be depositedon a flat, curved or rotating substrate[42,43]. Simultaneousdrying allows building up parts reaching millimeter thick-ness with only small variations from layer to layer. The min-imum thickness of the layers is controlled by the size of thesprayed droplets and may be less than 50�m. A German re-search group used wet powder spraying for the manufactureof pore-size gradient microfilters[44].

2.1.1.5. Slurry dipping and slip casting.If a porous bodyis sequentially dipped into slurries with varying powdercharacteristics, liquid drag into pores by capillary forcesleaves surface layers with a stepped gradient behind[45].The same principles are valid for sequential slip casting[46,47]. These methods have a high potential for series pro-duction if a limited number of layers fulfils the needs of theapplication.

2.1.1.6. Solid freeform processes.This class of computer-controlled, tool-less manufacturing methods can be readilyadapted to the production of FGMs. In the jet solidifica-tion process a hot powder-binder mixture with suitable flowproperties is deposited with an extrusion jet moved in twodimensions. During cooling the mixture solidifies and afree-standing body is formed. If the powder composition isvaried from layer to layer three dimensional graded bodiescan be formed. In a project of the priority program a two pis-ton construction with constant volume flow combined with astatic mixing chamber with a very small mixing volume fora volume flow rate of 32 mm3/s (Fig. 1) was developed forthis purpose[48,49]. A CAD model of a part was sliced intoindividual layers and a defined composition was associatedwith each layer. By changing the feedstock composition for

Fig. 1. Two piston construction for the multiphase jet solidificationequipment.


each layer gradients were produced. SiC–TiC was chosen asa non-reacting system. The viscosity of the feedstock (solidcontent: 47 vol.%) and the sintering behavior had to be ad-justed for varying compositions. With 8–10% liquid-phasesintering additives final densities of 97–99.5% were obtainedregardless of the SiC:TiC ratio. For smaller parts and thin-ner individual layers the three-dimensional printing processis more appropriate. In this process, each layer of the partis created by spreading a thin layer of powders and selec-tively joining the powder by ink-jet printing a binder mate-rial. After repeated operations the unbound loose powder isremoved to reveal a 3D green body. If the sprayed materialcontains a component that is not removed during the burnoutprocess, microstructural gradients can be manufactured by3D printing [50,51].

2.1.1.7. Other methods.The laser sintering process com-bines the deposition of powder layers and subsequentsintering[52] and can be modified into a stepped gradientprocessing method. Gravity or centrifugal sedimentation,electrophoretic deposition, pressure filtration, and centrifu-gal powder deposition may be used for the stepped type ofgradient structures, but have a bigger potential since they canbe applied to achieve continuous changes in the material.

2.1.2. Deposition of powders with continuouschanges in the mixture

2.1.2.1. Centrifugal powder forming (CPF) and impeller-dryblending. In CPF, powder mixtures with a continuousand computer-controlled change of the composition are fedonto a rotating distributor plate which accelerates them to-wards the inner wall of a rotating cylinder. A green bodyof sufficient strength is formed by simultaneously sprayingan organic binder onto the wall. The method is limited tocylindrical parts but offers a great flexibility in gradientdesign which is independent of the powder characteristics[53,54]. Centrifugal powder forming in combination withliquid-phase sintering was used in the priority program forthe production of W/Cu FGMs[55] with a ring diameter of120 mm, 40 mm height and up to 8 mm wall thickness. Atube like geometry may be produced by moving the formor the distributor plate along the axis of rotation during de-position. A related process is impeller-dry blending wheredry powders are mixed and homogenized in an impellerchamber prior to sedimentation in a mold[56]. No binderis required, and the process is restricted to flat FGM parts.

2.1.2.2. Gravity sedimentation.During sedimentation in acolumn differences in the particle velocity caused by differ-ent density or size of the powder particles lead to de-mixingof the different particle types[57–59]. As a result, a pore sizeor composition gradient is produced. If sedimentation occursin a liquid column free of particles a gradient with a continu-ous increase or decrease of the concentration of one particletype will be formed. If the sediment is directly formed from

the suspension a complicated transition function is retained.The bottom layer will still have the average composition ofthe suspension followed by an increase of de-mixing and thetop of the sediment contains only the powder fraction withthe lowest sedimentation velocity. A similar gradient will beobtained for a filtration process accompanied by sedimen-tation [60]. If sedimentation is carried out in highly loadedsuspensions, problems due to interparticle interactions occurfrequently. Another obvious disadvantage of sedimentationprocesses are limitations of the types of gradients that canbe produced. The laboratory equipment used in[58] was de-signed for small cylindrical FGM parts of 50 mm diameter.The sample height after the final consolidation was 10 mmor more. In spite of its limitations this method has a potentialfor the manufacture of tile-type FGM products with biggerdimensions due to its simplicity and reproducibility.

2.1.2.3. Centrifugal sedimentation.The formation of agraded structure by de-mixing of a powder suspension canbe achieved for finer particles using centrifugation[61].Due to the limited concentration in the suspension onlythin layers can be produced. Centrifugal sedimentation wasused in one project of the priority program aiming at thedevelopment of ceramic filtration elements with a pore-sizegradient. Pore-size graded filters of 18 mm diameter and athickness between 0.1 and 1 mm were made by centrifugaldeposition of TiO2 powders from aqueous suspensions con-taining a mixture of particle sizes between 10 nm and 5�m[62,63].

2.1.2.4. Electrophoretic deposition.Electrophoretic depo-sition from suspensions containing more than one compo-nent can be used to produce graded bodies. In the simplestcase an external mixing system supplies suspensions withthe variable concentrations of the components or the sec-ond component is added with time in calculated proportions[64–67]. The feasibility to produce a graded green body fromhomogeneous suspensions using the difference in the elec-trophoretic mobility of different powders was investigatedin the course of a priority program project[68]. The ve-locity of ceramic particles in suspensionsν depends on theelectrophoretic mobilityµ and the electric field strengthE,

ν = µE (1)

The mobility is defined by the properties of the dispersantand the zeta potential of the particlesζ:

µ = ε0εr

ηζ (2)

whereη is the viscosity andε0εr the dielectric constant ofthe dispersant. In a project of the priority program[68] thehigher mobility of Al2O3 compared to ZrO2 was used to pre-pare a gradient from a 2 vol.% suspension containing Al2O3and ZrO2. Electrolytic decomposition of the aqueous elec-trolyte and the resulting formation of gas bubbles in the de-posited layer were prevented by deposition of the particles on


a porous membrane. A proper choice of the surfactants andpH was necessary to guarantee stability of the suspensionand to achieve differences in electrophoretic mobility whichwere sufficient for de-mixing. Diluted suspensions were es-sential to prevent agglomeration and collective depositiondue to particle interactions at higher concentrations. Despitethe resulting longer deposition times graded deposits withseveral mm thickness could be produced. Due to the highermobility of Al2O3, it deposited faster and its concentrationin the suspension and deposit decreased with time. Accord-ingly, the zirconia concentration increased with time. Aftersintering, the gradient of the zirconia concentration was ac-companied by a grain size gradient because zirconia inhibitsthe grain growth of Al2O3.

2.1.2.5. Pressure filtration/vacuum slip casting.By con-tinuously changing the powder composition supplied to thefiltration system it is easy to obtain a defined one dimen-sional gradient in the deposit[69]. The same principles canbe applied to slip casting[47,70]. The aim of one priorityprogram project was the development of a SiC fuel evapo-rator tube for a gas turbine combustor. If the surface of theevaporator tube is porous a dramatic increase of the evapo-

Fig. 2. Design of the filtration apparatus (a) using controlled gas pressure; (b) using controlled liquid pressure.

ration rate can be achieved, however a direct transition fromporous to dense SiC leads to operating stresses exceedingthe strength of the SiC. FEM calculations[60] showed thata properly designed porosity gradient leads to a decrease ofthe failure probability at operating conditions by a factor of106 compared to a non-graded tube, if the permeability andevaporation rate are kept constant. Pressure filtration with avariable concentration of space holder particles was selectedto fabricate the required porosity gradients. Two experimen-tal set-ups, one using gas pressure (Fig. 2a) and one usingliquid pressure (Fig. 2b) were tested. A variation of spaceholder concentration from 0 to 40 vol.% could be achieved.After sintering the porosity in the tube was graded from 10to 50% over a distance of several millimeters (Fig. 3). Byusing segmented filter plates that allow for a local adjust-ment of the filtrate flow the production of axial and radialgradients is possible[71,72].

2.2. Sintering

The gradient that has been formed in the powder compactneeds to be preserved during sintering. In solid state sinter-ing of bulk specimens diffusion lengths are usually small


Fig. 3. Porosity gradient in a sintered ceramic sample produced by pressure filtration with PMMA beads as space holders.

enough to fulfil this condition. During liquid-phase sintering,however, macroscopic flow of the liquid phase can easilyoccur, and a gradient in the liquid phase is easily destroyed.Nevertheless, it has been demonstrated by Colin et al.[73]that it is possible to preserve concentration gradients dur-ing liquid-phase sintering of WC/Co hard metals. On theother hand, it is possible to make conscious use of liquidflow during liquid-phase sintering for gradient formation. Ina priority program project, this was studied extensively. Ifthe particle size of the solid component was graded, liquidredistribution by capillary forces occurred and produced aconcentration gradient[20]. The same effect was found ifthe proportion of cubic and hexagonal carbides is graded orif the carbon or nitrogen content varied in the green com-pact [20]. These redistribution processes are driven by thetendency of the system to minimize the free energy of in-ternal surfaces and interfaces. The gradient formation startswith melting of the liquid phase and is fast. Gradients inbinder content and hardness can extend up to several mil-limeters (Fig. 4). Distortions of the parts can be preventedby carefully adjusted green densities of the different regions.In hard metals the interaction of the gas atmosphere with thecompact during sintering produces chemical gradients with astrong influence on the phase content in a near-surface layer.Tough surface layers free of mixed carbides for steel cuttingtools have been obtained and studied[74–76]and the role ofnitrogen was described[22]. In WC/Co hard metals a car-bon gradient obtained by decarburizing results in a Co-richgraded surface layer[23,24]. A detailed study performed inthe priority program project showed a very complex behav-ior for the gradient formation[20]. A continuous change ofW, Ti, Zr, and Co or Ni content was obtained by slow diffu-sion controlled processes if carbon or nitrogen changes wereproduced by interactions with the sintering atmosphere. Itwas found that the cooling regime after liquid-phase sin-

tering is very important, and that a surface layer of up to60�m thickness enriched or depleted of binder metal couldbe adjusted.

2.3. Methods not requiring a sintering step

Coating techniques using powders for the materials sup-ply easily can be adopted to gradient fabrication by includinga mixing unit in the powder supply chain (e.g.[77,78]). An-other possibility is the use of two spraying guns with defineddeposition rate and motion of the guns (e.g.[79]). Feedingof mixed powders with controlled compositions is also thebasis of the laser cladding technique for building up gradedcoatings or graded freeform components[80–87]. Two

V24/8: WC0,6µm6Co WC11µm15Co

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10

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1 2 3 4 5 6 7 8 9 10

Distance [mm]

Co-

Con

tent

/Har

dnes

s/F

ract

ure

Tou

ghne

ss

Fig. 4. Co-concentration (�), hardness (�) and fracture toughness (�)gradient in a hard metals with a gradient of the carbide grain size. Linesare guides for the eye only.


Fig. 5. Left: generation of a metal–carbide composite with composi-tional gradient by laser beam cladding. Right: graded NiCrBSi tube with0–80 vol.% Cr3C2.

separately controlled pneumatic powder hoppers were usedto feed Ni-alloy and Cr3C2 powder simultaneously to deposittracks with a height 100–400�m and a width 400–800�mwith a 600 W Nd-YAG laser (Fig. 5). Subsequent claddingof overlapping tracks and deposition of tracks onto eachother with a changing ratio of the powder feed rates pro-duces a graded freeform component. The carbide contentwas increased from 0 to 80 vol.%. Laser and process pa-rameters have to be optimized to obtain crack and pore freesamples and to control the microstructure. For this studyNiBSi, NiCrBSi and NiCr50 alloys and fine (10–45�m)and coarse (45–106�m) Cr3C2 powder were tested. Asan example the hardness profile of a cylindrical part withwide change of the carbide content (fine grade) is shown in(Fig. 6). FEM calculations demonstrated that stress concen-trations at the interface with the carbide free substrate are

250

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650

750

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0 10 20 30 40 50 60 70 80 90Carbide Content [vol-%]

Har

dnes

s H

V0,

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NiBSi + Cr3C2NiCrBSi + Cr3C2NiCr50 + Cr3C2

Fig. 6. Hardness profile of graded NiBSi+ Cr3C2, NiCrBSi + Cr3C2

and NiCr50 + Cr3C2 cylinders obtained by laser beam cladding.

avoided and a thick wear resistant graded coating can bemanufactured by laser cladding. Other methods using pow-ders as starting materials or PM-related techniques that canbe modified for gradient processing are the Osprey-Processand centrifugal casting with a SHS-reaction.

In conclusion the disperse state of the materials in powdermetallurgy and ceramics processing offers a great variety ofmethods to obtain a graded microstructure or compositionin the final part. Most methods create a one dimensionalgradient, but developments in free form manufacturing andspecial deposition units open up the three dimensional de-sign of gradients. The main difficulties to produce gradedparts by powder technology arise from the consolidation ofthe inhomogeneous powder compact during sintering. Eachgraded system requires carefully selected powders and sin-tering conditions or the application of external pressure toreach full density without distortion. Some of the methodsdescribed are very cost-effective and have a good potentialfor industrial application. Others will be reserved for labo-ratory studies or special applications.

3. Melt processing

Gradient formation can be conveniently achieved by trans-port processes in the molten state and subsequent consoli-dation. Melt processing is widespread for FGMs containinga metal as one constituent.

3.1. Centrifugal casting

In centrifugal casting, particles of a refractory phase aredispersed in a metal melt. These particles may be formed insitu during cooling of the melt[88] or dispersed in a pre-ceding step. The density difference between particles andthe melt leads to the formation of a particle concentrationgradient if the melt is cast in a centrifuge. Particles witha lower density (Mg2Si, [88]) and a higher density (Al3Ti,[89]) than the melt have been used to prepare functionallygraded aluminum tubes which are selectively particle rein-forced at the inner or outer surface of the tube. If plateletsare used as particles a gradient in particle orientation can beproduced during the casting process[90].

3.2. Sedimentation casting

In the framework of the priority program, gradient forma-tion by sedimentation of particles was investigated[91]. Thegoal of the project was the production of a one dimensionalFGM with an acoustic impedance gradient (the acousticimpedance is defined as the product of density and soundvelocity). Impedance graded FGMs can reduce the acous-tic mismatch between a high impedance PZT ultrasoundtransducer and a low impedance material such as bodytissue. The impedance gradient material was produced bydispersing WC particles in a Cu/Mn alloy and subsequent


Fig. 7. Micrograph of a WC/Cu/Mn alloy gradient and the corresponding spatial distribution of the WC particle concentration.

sedimentation. In this way a particle concentration gradi-ent from 0 to 60% WC was produced (Fig. 7). The highimpedance WC side matched the impedance of PZT ex-actly. Efforts to produce W/Al and W/Sn FGMs with thismethod failed due to solubility problems (W/Al) or poorwetting of the particles with the melt (W/Sn).

3.3. Controlled mold filling

In a further project of the priority program, the function-ally graded material was produced by successive casting oftwo melts[92]. In a gravity casting process, the mold waspartially filled with the first melt, and in a further step, thesecond melt was cast on the partially solidified first ma-terial (Fig. 8). Gradient formation occurred by forced andthermal convection, and the width of the graded interfacewas predominantly controlled by the degree of solidifica-tion of the first melt at the time when the second melt wascast. In a modification of this process casting was carriedout in a rotating mold, and components with a cylindricalshape could be obtained. The controlled casting process wasused to produce AlSi7/AlSi18 and A356/Duralcan gradedmaterials. In a thermal shock experiment, graded and un-

Fig. 8. Multi-alloy gravity casting process.

graded A356/Duralcan composites were heated to 400◦Cand quenched in water. After 700 thermal cycles a gradedspecimen showed no damage (Fig. 9a). In contrast, an un-graded specimen displayed cracks already after 300 cycleswhich eventually lead to a complete destruction of the spec-imen (Fig. 9b).

Fig. 9. A356/Duralcan composites after 700 thermal cycles at 400◦C: (a)specimen with graded interface; (b) specimen with ungraded interface.


Fig. 10. Set-up for directional solidification experiments with natural convection (a) and forced convection (b).

3.4. Directional solidification

The possibilities of generating a macroscopic, one-dimensional concentration gradient with radial symmetryby directional solidification were also investigated in theframework of the priority program. In this process, the dif-ference between the solidus and liquidus composition dur-ing directional solidification of an alloy is used to preparea FGM. In order to avoid solidification under steady statewith constant solid concentration, the melt in front of thesolidification front must be stirred as effectively as possible.This can be achieved by natural convection in a verticalresistance furnace (Fig. 10a). A temperature gradient wasgenerated by inserting cooling coils in the top end of thefurnace. Samples were melted inside the furnace and thenmoved toward the chilled zone with a constant velocity. Thesolidification direction was thus upwards, and for alloyingelements with higher density than the matrix element, a so-lutally and thermally unstable density gradient in the meltwas generated. For alloying elements with similar density as

Fig. 11. Concentration distributions in Al-4 wt.%Cu alloys, solidified with (�) 0.9�m/s, (�) 9.0�m/s, (�) 46�m/s, (�) 264�m/s velocity in aninduction furnace.

the matrix element, an induction furnace was used. A largetemperature gradient was established by submerging the tipof the sample in cooling water causing forced convectionin the melt (Fig. 10b). Using natural convection, completemixing could not be achieved during directional solidifi-cation of an Al-4 wt.%Cu alloy even at the lowest samplevelocity of 0.34�m/s. Better mixing could be realized withforced convection in the induction furnace for specimenswith 8 mm diameter (Fig. 11). The minimum concentrationat a velocity of 0.9�m/s was 0.56 wt.%, which is closeto the theoretical value of 0.48 wt.% for complete mixing.For this velocity, a Scheil type concentration profile withthe maximum possible degree of segregation was obtained.The main reason for the higher stirring efficiency in theinduction furnace was the radial temperature gradient intro-duced by the preferential heating of the sample surface. Thedegree of segregation in the induction furnace decreasedwith increasing sample velocity and segregation-free mi-crostructures were generated at velocities above 0.3 mm/s(Fig. 11).


3.5. Infiltration processing

Infiltration is a suitable processing method for function-ally graded materials containing phases of very differentmelting points. In this process a preform of the more re-fractory phase possessing a porosity gradient is producedand infiltrated with the melt of the lower melting compo-nent at elevated temperatures. The preform must containonly open pores and must be insoluble in the melt. Infiltra-tion processing is particularly attractive for metal/ceramicand glass/ceramic FGMs. However, metal/ceramic systemsare generally non-wetting (with a wetting angleΘ > 90◦)and the preforms are not infiltrated spontaneously. Largerpores can be infiltrated by applying pressure during infiltra-tion. The required pressurep for a complete infiltration of acylindrical pore of radiusrp is given by

p = −2γLV cosΘ

rp(3)

whereγLV is the surface tension of the melt. For a cylin-drical pore in a ceramic/metal system with a pore radiusof 5�m, a wetting angle of 120◦ and a surface tension of1 J/m2 a pressure of 0.2 MPa is necessary for infiltration.Pressures required for complete infiltration of real pore struc-tures are higher, but can be reached using squeeze casting orgas-pressure sintering equipment. An alternative to pressureapplication is the improvement of the wettability by precoat-ing the preform, by suitably alloying the melt or changingthe composition of the gas atmosphere. However, even for awetting angle of about 30◦ a pressure of about 1 atm is nec-essary for complete infiltration of a particulate preform[93].Further limitations of infiltration processing are caused bythe restricted availability of graded porous preforms whichmust not contain closed porosity and need to have a suffi-cient mechanical strength at high porosity in order to resistthe pressure applied during infiltration.

One processing method for such preforms is based onpowder compacts with a graded particle size. During sin-tering, the particle size gradient leads to the formation of aporosity gradient[94]. Since this is accompanied by distor-tion, shape stabilization during sintering is required. Othermethods which have been used to produce graded preformsinclude sintering of green bodies with a graded content ofshort fibres[95], transformation of a Al/Al2O3 gradient intoa porosity gradient during sintering[96], transformation ofa gradient in carbon content in TiCx into a porosity gradientduring sintering[97], plasma spraying with varying processparameters during coating deposition[98], coating of a poly-mer foam with ceramic slip in a centrifuge and burn-out ofthe polymer[99], and solid freeform fabrication[100].

In a project of the priority program a combination of cen-trifugal powder forming and liquid-phase sintering was usedto prepare a porosity gradient. For this purpose the amountof liquid forming Cu was graded in a W/Cu green body bycentrifugal powder forming, and the movement of the liq-uid Cu towards a more homogeneous distribution caused a

Fig. 12. W/Cu gradient formation: (a) starting condition after centrifu-gal forming; (b) porosity gradient after liquid-phase sintering; (c) W/Cugradient after Cu-infiltration.

porosity gradient in the W/Cu composite[54,55]. Infiltra-tion of the pre-sintered body with additional Cu leads to afully dense graded W/Cu composite (Fig. 12).

Electrochemical dissolution of porous precursor materialsas a route to produce graded preforms has been investigatedin a different project of the priority program. The experi-mental set-up consists of a simple electrolysis cell with aporous anode material that undergoes dissolution (Fig. 13).If a current passes through the cell, potential gradients insidethe porous anode develop and lead to a local variation in the

porous anode cathode

x

yz

0 L

contact contactelectrolyte

Fig. 13. Sketch of the experimental configuration for electrochemicalgradation.


Fig. 14. Microstructure of a graded W/Cu composite produced by elec-trochemical gradation.

rate of dissolution and porosity of the anode. A quantita-tive description of the porosity evolution was possible by anelectrokinetic model described inSection 5. Electrochemi-cal gradation has been used to introduce porosity gradientsin materials such as bronze[101] and copper[102]. Porositygraded tungsten preforms also have been produced by thisprocess[103]. After anodization the tungsten preforms wereinfiltrated with molten copper applying 10 MPa Ar pressureto yield a fully dense W/Cu FGM (Fig. 14).

Electrochemical gradation was also used to produceporosity graded ceramic preforms. In this case porous car-bon was used as an anode material and sacrificial preform.After electrochemical gradation and infiltration with analumina slip of sufficiently small particle size, the carbonwas burnt out[104]. The retained ceramic preform was in-filtrated with Al to produce a Al/Al2O3 functionally gradedmaterial. A very versatile method to produce porosity gradedsacrificial preforms is also the inhomogeneous compactionof polymer foams as first described by Cichocki et al.[105].In this so-called “gradient materials by foam compaction”(GMFC) process open-celled polymer foam of a low initialdensity is used as a starting material. Porosity graded foamsare produced by warm pressing foam wedges of differentshape to a homogeneous thickness or by stacking com-pacted foam plates with different compaction ratios. Withboth methods a well controlled macroscopic porosity distri-bution inside the foam with minimum values of 40% openporosity and maximum values of 97% open porosity can be

Fig. 16. Schematic diagram showing the impregnation process used for the introduction of spacer particles into the fiber bundle.

Fig. 15. Manufacture of a Cu/Al2O3 FGM by the GMFC process (process-ing stages are numbered consecutively): (1) uncompressed polymer foam;(2) foam plates compressed with different compaction ratios; (3) stackedfoam plates; (4) foam stack infiltrated with Al2O3; (5) graded Al2O3 pre-form after sintering; (6) preform material cut to specimen dimensions;(7) retained Cu/Al2O3 FGM after infiltration of the preform with moltenCu; (8) specimen for burner rig test after grinding and polishing.

produced[106,107]. The compacted foams are infiltratedwith a ceramic suspension. After water removal by filtrationand drying, the polymer foam is burnt out and the ceramicis sintered. The retained porosity graded ceramic preformcan be infiltrated with a metal to produce a metal/ceramicFGM. Fig. 15illustrates the processing steps on the exampleof a graded Cu/Al2O3 material for burner rig testing[108].

In a further priority program project, fiber preforms withgraded fiber spacing were produced by impregnating car-bon fibers with slurry[109]. The principle of this method isto introduce ceramic particles into the fiber bundle therebyacting as spacers.Fig. 16shows an illustration of the usedimpregnation set-up. It was found that the suspension con-centration and the winding speed of the fibers are the mostimportant parameters to determine the fiber volume contentof the preform (Fig. 17). Higher particle concentrations andlower winding speeds result in a distinctly more pronouncedaccumulation of spacer particles between the fiber filamentsand consequently lower fiber volume fractions. After infil-tration of the fiber preforms with different magnesium al-loys carbon fiber reinforced composites with graded fibercontent were obtained. The volume content of the reinforc-ing fibers could be continuously varied between 25 and 60%


Fig. 17. Influence of the winding speed on the fiber volume content for several Al2O3 particle concentrations. The fiber content increases with higherwinding speeds, whereas the amount of spacer particles in the fiber bundle decreases.

by adjusting the winding speed during the impregnationprocess.

3.6. Reactive infiltration

In reactive infiltration the preform reacts with the infil-trated material to form a graded composite. The advantagesof reactive infiltration are low-cost processes and raw mate-rials and improved wetting. The gradient may be formed byincomplete infiltration of a homogeneous porous body or bycomplete infiltration of a preform that contains a porosityor composition gradient. Partial infiltration is usually com-bined with a reaction and sintering step after infiltration inwhich a dense body containing the new phases is formed.Mullite/alumina [110], mullite/aluminumtitanate/alumina[111] and alumina/calcium-hexaluminate[112] FGMs wereformed by partially infiltrating precursors such as silica solsor calcium acetate into alumina preforms. Partial infiltrationis a very simple method to produce continuous gradients,but lacks flexibility in the shape of the composition profile.If complete reactive infiltration is used to produce gradients,the composition profile of the product can be controlled bythe gradient in the preform. An example is the so-calledi-3A process[113], in which Al is infiltrated into porouspreforms consisting of Al2O3 and a metal oxide. The metaloxide is reduced by the liquid aluminum, and Al2O3 andthe respective metal aluminides are formed according to thereaction

aMeuOv + bAl → cAl2O3 + dMexAl y (4)

For the production of FGMs by the i-3A process gradedAl2O3/TiO2 and Al2O3/Fe2O3 bodies were produced bypowder stacking or slip casting. After gas-pressure infiltra-tion with Al, the graded green bodies were converted intoAl2O3/TiAl 3 and Al2O3/FeAlx FGMs[114]. The aluminidecontent in the FGMs varied between 0 and 35 vol.% and haddefect-free interfaces between layers of different composi-tion.

4. Processing of polymer-based FGMs

In polymers, like in other materials, compositional andmicrostructural gradients are intended to allow an opti-mum combination of component properties, for exampleweight, surface hardness, wear resistance, impact resis-tance and toughness. Polymers with a porosity gradient,so-called polyurethane integral skin foams provide high im-pact strength at low weight and have been used since longfor instrument panels or head rests in cars[115]. Gradedpolymers that have been processed so far include gradedfiber composites[116], graded interpenetrating polymernetworks[117], graded biodegradable polyesters[118] andgraded index polymer fibers and microlenses[119]. Com-pared to ceramic or metal based systems, the knowledge onprocessing methods for polymer FGMs is limited. In thefollowing, some important processing methods for polymerFGMs are described where particular attention is given tothe methods used in the priority program.

In polymer composites, a gradient of the reinforcementcan be introduced by centrifuging prior to polymerization[120]. This method was studied in detail by Klingshirn et al.in a priority program project[121]. Two resins were usedas matrix materials, while SiC particles, aramide particles,glass fibers and carbon fibers were used as filler materials.Rings and tubes with a gradient in filler content were pro-duced by centrifuging a dispersion of particles in a resinprior to hardening. The average filler content was chosenbetween 5 and 45%. After centrifugation the filler content inthe outer portions of the rings increased up to 27% for glassfibers and to 45% for SiC particles, for an average filler con-tent of 20%. This significant difference between differentfiller materials was primarily attributed to the aspect ratio ofthe filler particles, which was about 200 for the glass fibersand 1 for the SiC particles. A smooth gradient in the fillerconcentration could be produced by adjusting the rotationspeed of the centrifuge and the viscosity of the resin. Thelatter could be increased by gelation for a given time before


centrifugation. A viscosity 25 Pa s. was found suitable forcentrifugation of carbon fibers—a lower viscosity of 5 Pa slead to a complete sedimentation of the carbon fibers duringthermosetting. The wear resistance of thermosetting poly-mers was improved by the addition of the hard filler parti-cles or fibers. It was demonstrated that the gradient in fillercontent leads to a corresponding gradient in microhardnessand wear resistance[122,123].

Long-fiber reinforced polymers with graded fiber com-position or orientation can be produced by lamination tech-niques[124]. In a priority program project a gradient wasintroduced also in the matrix material. Polycyanurate resinswere used as a matrix material. Since the neat polycyanurateis too brittle for many structural applications, it was modifiedby a rubber modifier. Higher modifier contents improve thetoughness, but the good high-temperature stability of neatpolycyanurate and the resistance against water absorptionare reduced. The idea of the project was thus to optimizethe properties of a component by a gradient in modifiercontent, in addition to gradients in fiber density and orienta-tion which are already used in lightweight components. Thegradient was achieved by hot press lamination of a stack ofglass fiber prepregs containing differently modified resins.Since the resins are redistributed between the prepreg layersduring molding, it was necessary to determine the result-ing gradient at the end of the process. This was done bydynamic mechanical analysis (DMA) which proved that asmooth gradient was established during molding[125].

If polymers develop distinctly different microstructures atdifferent processing temperatures, it is possible to introduceproperty gradients by applying a temperature gradient dur-ing processing. Such an effect was used in a priority programproject aiming at the development of a polymeric impact pro-tection material consisting of a single component. Sinteredfiber materials were selected as candidates because they havehigh energy absorption potential. The properties of the fi-nal product depend strongly on the temperature at which thefibers are sintered together. Low sintering temperatures leadto an anisotropic microstructure, in which the aligned fibersare still well discernible. High sintering temperatures leadto a compact, isotropic material. Components with either ofthese microstructures have poor to moderate impact resis-tance. The anisotropic material fractures by delamination be-tween the fibers, so that small objects can penetrate through

Fig. 18. Graded microstructure of sintered Vectran® fibres.

the material between the fibers and the high strength of thefibers cannot be used for the absorption of impact energy. Onthe other hand, the isotropic microstructure, per se, has lowimpact resistance. The idea of the project was to introduce amicrostructural gradient by applying a temperature gradientduring sintering with the goal to improve the impact prop-erties of the component. For one of the test materials, theliquid-crystalline copolyester Vectran®, the idea was suc-cessful.Fig. 18shows an example of a graded microstructurewith a nearly homogeneous structure at the high-temperatureside and clearly discernible fibers at the low-temperatureside. The impact resistance was measured in a repetitivedrop-weight test, in which the plates are impacted repeat-edly with increasing impact energy until penetration occurs.In this test, the homogeneous isotropic material exhibitedvery low energy absorption values around 20 J, while thehomogeneous material with anisotropic structure (sinteringtemperature 290◦C) gave energy absorption values around200 J with a small difference between cross-ply and fabric.The best values, nearly 300 J, were obtained with the gradedmaterial, but only when the anisotropic microstructure wasat the impact side and the isotropic structure is at the oppo-site side, and cross-ply rather than fabric was used[126].

5. Modeling of FGM processing

The rapidly increasing performance of computers andcomputer software has favored the widespread use of nu-merical simulation tools for the design of parts and for theoptimization of manufacturing processes. Process modelingmay be especially important in connection with function-ally graded materials, since gradients often cause specialproblems. In some cases the simulation can help to estab-lish the desired gradient, and to maintain it throughout theentire process by an appropriate process control. In othercases, process modeling may help to circumvent or at leastminimize problems such as warpage or cracking during thefabrication of a component.

5.1. Modeling of gradient formation

In many processes such as powder stacking the gradient isformed directly and there is no need to model the formation


of the gradient. When the powders are not deposited directlybut in a filtration process knowledge of the rate of cake for-mation is necessary to produce pre-defined gradients. It hasbeen demonstrated by Tuffé and Marple that a precise con-trol of gradient formation is possible using a simple analysisif the casting parameter is known for each slip composition[127]. As long as particle interactions are small, modelingof particle sedimentation is straightforward if the particlesize distributions are known. Normally, the Stokes equationis used to describe the particle velocity, but for heavier par-ticles Newton’s law has been used[58]. Reasonable agree-ment of calculated and observed particle volume fractionshave been achieved for sedimentation experiments[59,128].Modeling of gradient formation has also been carried outfor centrifugal casting[129,130]and reasonable agreementwith the experimental particle distribution was found[130].In vapor deposition processes the gradient is usually intro-duced by dosage of precursors from separate containers orby adjusting the partial pressure of a reactive gas. For suchprocesses the relationship between composition of the gasphase and the composition of the deposit may be compli-cated and knowledge of the relationship between the partialpressures of different components in the gas phase and thefilm growth rate and composition is required for the designof controlled gradients[131].

Polyurethane integral skin foam parts exhibit a substan-tial density gradient, such that the surface is relatively denseand mechanically stable, while the core of the part has lowdensity. This is achieved by keeping the tool surfaces rela-tively cool (typically 80◦C), whereas the center of the part isheated by the foaming reaction to higher temperatures (typ-ically 180◦C). This leads to an expansion of the foam bub-bles to much larger diameters, which means lower density.In a priority program project the foaming process was sim-ulated numerically using the finite-element code ABAQUS.Predicted and optically measured density distributions wereconsistent, asFig. 19 shows[132]. Further, a new methodwas developed to detect the gelation of the foam material byultrasound transmission and a suggestion was made how agraded foam structure could serve as a coupling element forultrasound between air and a sensor, and how the elementcould be fabricated.

In the previous chapter, a priority program project usinganodic dissolution to prepare graded porous preforms for in-filtration processing was described. In the following a modelis presented that allows an accurate prediction of the poros-ity gradient developing during the electrolysis due to the dis-solution of the electrode material. A continuum model wasused which regards the porous electrode as a macrohomoge-nous mixture of the electrode matrix and the electrolyte fill-ing the pores. It was assumed that the local density of theFaraday current per unit internal surface areajF is governedby the Tafel equation:

jF(η) = j0 exp

(αF

RTη

)(5)

wherej0 andα are constants for a given electrode reactionandη the overpotential (driving the electrode reaction). Fora plate-like porous anode of thicknessL where the currentfeeder is located at the side opposite to the cathode (Fig. 13)the current density per unit volumejv (and thus the localrate of dissolution) could be determined analytically as:

jv = SjF = j0SB

2A

{1 + tg2

([w− wmin]

√B

2

)}(6)

where the relative positionw = x/L and the surface areaper unit volumeShave been introduced.A andB are dimen-sionless constants defined by the following equations:

A = (ρ∗l + ρ∗

s)L2Sj0

αF

RT(7)

arctg

(C√B

)+ arctg

(D√B

)=

√B

2(8)

with

C = ρ∗sjTL

αF

RTand D = ρ∗

l jTLαF

RT(9)

The position of minimum overpotential is determined by thedimensionless quantitywmin:

wmin =(

2√B

)arctg

(C√B

)(10)

ρ∗s and ρ∗

l are the effective resistivity of the porous elec-trode and electrolyte, which depend on the pore structureof the electrode and the resistivity of the employed materi-als. Eq. (6) is only valid if the current potential relation isdescribed by the Tafel relation. A finite-difference schemebased on the same differential equations as the analytic so-lution was developed in order to be able to use arbitrarycurrent–potential relations. As material is removed by theanodic reaction a position-dependent porosity and specificsurface area develops. This was also included in the nu-merical model by the dividing the operation time into smalltime intervals and adding of the material removal duringall time intervals. The results of the analytical and the nu-merical model were compared to experimental data for aporous tungsten electrode (Fig. 20). Already the analyticalmodel showed a good agreement with experimental data, butthe numerical model could predict the porosity distributioneven with a higher precision. The produced porosity distri-bution can be controlled by a number of experimental pa-rameters (Fig. 21). Increasing the total current and keepingthe other experimental parameters constant lead to a moreuneven tungsten distribution (compare a and b). Increasingthe resistivity of the electrolyte by decreasing the concen-tration of electrolyte also leads to a more uneven tungstendistribution (compare c and d). Preforms with lower initialporosity graded under the same experimental conditions alsoshowed more uneven tungsten content (compare b and d).Porosity distributions produced by the electrochemical gra-dation process were thus predicted quantitatively in a nu-merical model of the gradation process taking into account


distance from upper surface [mm]

0 2 4 6 8 10

dens

ity [k

g/m

³]

average density

300

400

600

700

500

800

900

294 kg/m³376 kg/m³480 kg/m³597 kg/m³680 kg/m³

experimentsimulation

Fig. 19. Measured and calculated density distributions across the thickness of a polyurethane plate after the foaming reaction.

microstructural changes of the porous electrode during thedissolution process.

5.2. Modeling of sintering

As a basis for describing the sintering of graded compo-nents, models for solid-state and liquid-phase sintering weredeveloped and applied in the priority program[133–135].The models were implemented in a finite-element code(ABAQUS), and they were also combined with beam-theoryand plate-theory PC programs in order to predict thewarpage and cracking of graded components. Finally a pro-gram for microwave sintering was developed and combined

Fig. 20. Comparison of the predicted and observed tungsten content inan electrochemically graded W/Cu composite, (- - -) analytical model;(—) numerical model. The numerical model includes the effects of struc-tural changes on the current distribution in the porous electrode. Initialtungsten content was 75 vol.%, current density 12.2 mA/cm2 and NaOHconcentration 2 mol/l.

with an advanced solid-state sintering model. The modelsand their application in the priority program are describedin the following paragraphs.

5.2.1. Application of a liquid-phase sinteringmodel to Al2O3

In [134,135] a liquid-phase sintering model was devel-oped, which was used in this project to describe the sinteringbehavior of 94% purity Al2O3. To determine the parame-ters of the model, so-called sinter forging tests were carriedout. In this test, a cylindrical specimen is subjected to anaxial load during sintering, and the axial and radial strainsare measured. In addition the evolution of the microstruc-ture was observed, since the models contain the grain sizeas an internal state variable. The parameter fit is done us-ing a PC program by integrating the evolution equations ofthe model and varying the parameters by hand, until thesimulated behavior agrees with the measured one.Figs. 22and 23demonstrate that the evolution of the strains (andhence of the density) and of the grain size can be describedwell by the model.Table 1shows the resulting parametervalues. It should be noted that a similarly good fit of the datacan be achieved by a solid-state sintering model[133]. Thisis not surprising, since for relatively low liquid contents thestructure of the liquid-phase model is similar to that of thesolid-state model.

5.2.2. Warpage of graded platesThe warpage of graded plates during sintering was mod-

eled in two ways. First a relatively simple program waswritten in the framework of plate-theory. The equations forthe equilibrium of force and bending moment were given in[136]. Second, the sinter warpage of a plate with a grain sizegradient was calculated with a finite-element code[62,136].

As an example, a circular plate with 10 mm thickness and100 mm diameter is considered in[136]. The lower half ofthe plate consists of molybdenum and the upper half of ZrO2.A linear heating rate of 5 K/min and a hold temperatureof 2173 K are prescribed with a linear temperature gradientof 100 K over the plate thickness during the rise time, the


VIC

KE

R'S

HA

RD

NE

SS

[GP

a]

POSITION [mm]

80

80 80

80

20

20 20

20

00

0 0

3

3 3

0 2 4 6

0 2 4 6

0 2 4 6

0 2 4 6

TU

NG

ST

EN

CO

NT

EN

T [V

ol.-

%]

TU

NG

ST

EN

CO

NT

EN

T [V

ol.-

%]

TU

NG

ST

EN

CO

NT

EN

T [V

ol.-

%]

VIC

KE

R'S

HA

RD

NE

SS

[GP

a]

TU

NG

ST

EN

CO

NT

EN

T [V

ol.-

%]

2 2

3

22

1 1

11

60

40

6060

4040

60

40

(a) (c)

(b) (d)

Fig. 21. Comparison of the predicted (—) and observed (�) tungsten distribution in electrochemically graded W/Cu for different current density, NaOHconcentration and initial porosity. Open triangles are hardness values of the composites: (a) initial porosity 25%, current density 5.5 mA/cm2, cNaOH

2 mol/l; (b) initial porosity 25%, current density 12.2 mA/cm2, cNaOH 2 mol/l; (c) initial porosity 42%, current density 5.5 mA/cm2, cNaOH 2 mol/l; (d)initial porosity 42%, current density 5.5 mA/cm2, cNaOH 1 mol/l.

Table 1Parameters of the liquid-phase sintering model for 94% purity alumina

Initial values Relative green densityD0 0.605Initial grain radiusR0 1.05�m

Melt formation Start of melting 1430 KMelting rate 0.000595 KMaximum volume fraction of melt 0.1

Grain rearrangement Maximum solid density by rearrangement 0.63Viscosity, pre-exponential factor 1× 105

Viscosity, activation energy 425 kJ/mol

Solution/re-precipitation Reaction rate, pre-exponential factor 3.9× 104

Reaction rate, activation energy 400 kJ/mol

Diffusion in liquid phase Diffusion coefficient, pre-exponential factor 5.2× 1038

Diffusion coefficient, activation energy 425 kJ/mol

Sintering stress Solid/liquid interface energy 0.25 J/m2

Liquid surface energy 0.25 J/m2

Volume fraction of large pores 0.1Parameter for size distribution of large poresB 0.12Parameter for size distribution of large poresn 1

Ratio shear-to-bulk viscosity G/K 0.27


200 250 300 350 400-0.5

-0.4

-0.3

-0.2

-0.1

0.0

40 N45 N

35 Naxial strain

radial strain

free sintering

Str

ain

Time in min

Fig. 22. Evolution of axial and radial strains. Solid lines: experiments;dashed lines: model. Temperature is the same as inFig. 23.

ZrO2 side being cooler. The material parameters that aremost important for the sintering behavior of the plate are theactivation energies for grain boundary diffusion,Qb, and forgrain boundary migration,Qm. They are taken from otherinvestigations to beQb = 280 kJ/mol for Mo, 271 kJ/mol forZrO2 andQm = 230 kJ/mol for Mo, 310 kJ/mol for ZrO2.The relative green densities are assumed to be 0.6 for Moand 0.5 for ZrO2. The initial grain radius is chosen as 2.5�mfor Mo, and that of ZrO2 is varied (0.1, 0.5 and 1�m).

Fig. 24shows the calculated evolution of the warpage ofthe plate (the normal displacement of the edge of the platewith the center being fixed). The fine-grained ZrO2 sintersat a temperature where Mo is still practically inactive. Thisleads to a displacement of the plate edge in upward direction.Later also the Mo starts to sinter, but it does not nearly reachfull density, since intense grain coarsening prevents furtherdensification. Hence the plate is substantially distorted at theend. This distortion can be minimized by choosing the ap-propriate grain size for the ZrO2 powder. If the grain radiusof the ZrO2 powder is 1�m the Mo side starts to sinter first,but slightly later also the ZrO2 shrinks and just compensatesthe initial warpage. In this case the final relative densitiesare 0.74 in ZrO2 and 0.88 in Mo.

200 250 300 350 4002.0

2.5

3.0

3.5

Experiment

Model

Al2O

3

A1894

Gra

in S

ize

in µ

m

Time in s

1000

1100

1200

1300

1400

1500

1600

1700

Tem

pera

ture

in ˚

C

Temperature

Fig. 23. Evolution of the grain size. Symbols: experiments; dashed line:model.

0 2 4 6 8 10 12 14-10

0

10

20

30

40

R= 1 µm

R= 0.5 µm

Grain radius of ZrO2 R= 0.1 µm

Mo-ZrO 2∆T = 100 K

Dis

plac

emen

t in

mm

Time in h

Fig. 24. Warpage of a plate consisting of molybdenum and zirconia. Thefinal warpage is small, if the zirconia has a grain radius of 1�m.

Within the framework of plate-theory the distortionnecessarily remains axisymmetric. A three-dimensionalfinite-element calculation, however, shows that the initialsymmetry is spontaneously broken. The deformation isonly axisymmetric for small deflections, whereas the plateassumes the form of a potato chip as sintering progresses[136]. These predictions are consistent with observations ofMoritz et al. when sintering pore-size graded titania plates[62]. The sintering temperature was selected such that theplates acquire enough strength, but still have an open porestructure, even in the fine-grained top layers. It was foundthat up to 800◦C, practically no sintering occurred. Atabout 850◦C the plates developed a certain mechanicalstrength, and the pores were still open with pore sizes be-tween 5 and 50 nm at the fine-grained side, and 3�m atthe coarse-grained side (as measured by quantitative imageanalysis). With increasing sintering temperature the plateswere distorted, as the fine-grained side sintered faster thanthe coarse-grained side (Fig. 25). Part of the warpage wasreversed when also the coarse-grained material sintered inthe temperature range between 900 and 1200◦C. The sinterwarpage could be avoided without suppressing the radialshrinkage by covering the graded plate with an aluminadisc of about 3 g mass.

5.2.3. Prediction of crackingWhen graded materials are processed, cracking is a poten-

tial danger, since different regions sinter at different rates.The resulting incompatibilities may either be relieved bycreep accommodation and distortions, or they may lead tocracking, depending on the material properties and the geo-metrical conditions.

It is suggested that cracking is initiated in a sintering bodyif the mean stress locally exceeds the sintering stress. In thiscase the material starts to desinter, which means that it lo-cally softens. Softening may lead to strain localization; theincompatibilities between areas sintering at different rates


Fig. 25. Influence of the sintering temperature on the warpage of graded titania plates.

are accommodated in a narrow desintering layer, i.e. in theevolving crack. To illustrate the principle, the configura-tion shown inFig. 26 is considered: A spherical core offine-grained material (region 1) is separated by a thin layer(region 2) from the coarse-grained bulk (region 3). Ten-sile stresses develop between the coarse-grained bulk andthe fine-grained core due to the different sintering stresses.These stresses act on the interface layer, which is intended

Fig. 26. (A) Configuration analyzed in the sintering model. (B) Evolutionof the relative density in the fine-grained core, in the interface layer andin the coarse-grained bulk. Grain size in the interface: (a) 0.518�m; (b)0.52�m; (c) 0.522�m.

to serve as a perturbation to initiate cracking. To be specificthe grain size in the bulk is assumed to be 2�m, in the coreit is 0.5�m and in the interface layer it is given a value be-tween 0.5 and 2�m. The initial relative density is chosenas 0.55 except in the interface layer, where it is 0.54. Thequestion to be answered is whether the greater sinter activityof the fine-grained core leads to a decohesion from the bulkor whether the core remains attached, its sintering rate be-ing constrained by the slowly sintering bulk.Fig. 26showsthe result of the calculation for three different grain sizes inthe interface layer (cases a, b and c). The numerical valuesfor the grain sizes are deliberately chosen to illustrate thesudden transition from stable sintering of the system to un-stable desintering (cracking) of the interface. In case (a) thegrain size in the interface layer is 0.518�m. After a periodof retarded densification, the interface, as well as the com-plete system densify readily. In case (b) the grain size in theinterface layer is 0.52�m. Still the interface remains intactand reaches full density after a period of initial desinter-ing. In case (c), however, whered = 0.522�m, desinteringleads to a complete loss of density and strength of the inter-face layer. When the interface layer breaks down, the corestarts to sinter much faster than before when it was still con-strained by the slowly sintering, coarse-grained bulk. Hence,near the instability point, a seemingly minor change of thegrain size in the interface leads to a completely differentresult.

In further calculations, the grain size of the interface layerwas varied, while all other parameters remained fixed. Itturned out that stable sintering is only possible, if the grainsize in the interface is close to either that of the core (0.5�m)or that of the bulk (2�m). If the interface has grain sizes be-tween 0.52 and 1.95�m, decohesion occurs. Hence in thiscase, a transition layer with intermediate properties betweenthe core and the bulk is less favorable for the sinterabilitythan a sudden transition. This example is intended to illus-trate the principle how cracking can be described as a pro-cess of localized desintering for a simple configuration. Inthe framework of the finite-element method, the principlecan be generalized to arbitrary geometrical configurationswith property gradients.


5.2.4. Microwave sinteringMicrowave sintering offers certain advantages compared

to conventional heating, for example for the sintering offine-grained ceramic powders, when the fine grain size isto be preserved during sintering. On the other hand, theelectromagnetic field in the microwave cavity is inhomoge-neous, which is an additional difficulty for the sintering ofgraded parts. Hence a finite-difference microwave programwas written and combined with the solid-state sinteringmodel described in[133]. Commercial microwave codesgenerally solve Maxwell’s equations, sometimes coupledwith the heat conduction equation, since the dielectricproperties of materials usually depend on temperature. Inaddition to the temperature dependence, the thermal andelectric properties may depend strongly on the sinter den-sity, and the program developed here takes that also intoaccount[137].

The following example shows that the full coupling of thesintering model with the electromagnetic and thermal fieldsis not a second-order effect, but it may be critical for thesuccess of a sintering process.Fig. 27 shows the axisym-metric microwave furnace to be analyzed. The energy is fedinto the cavity by the bottom antenna, while the antenna atthe top serves to control the process. The material to be sin-tered is an alumina ceramic. The susceptors are assumed tobe dense alumina, and the insulation is given the thermaland electric properties of alumina fibers. The microwave fre-quency is chosen such that the TE031 mode is excited (forthe microwave terminology see[138]).

Fig. 28 shows the temperature distribution for the casethat a coarse-grained alumina with grain size 2�m isheated with 1 kW microwave power. The picture shows thata so-called runaway instability occurs: the temperature israther low everywhere in the ceramic (the hatched area inthe center) except in a small region, where it quickly risesto very high values. In the case shown here the temperaturereaches the melting point of alumina in the hot spot, whilethe rest of the ceramic does not even start to sinter. Such arunaway instability can occur if the microwave absorptioncoefficient increases with temperature, if the microwavepower is high, if the heat conductivity is low, and if thespecimen is large. Apparently the conditions for a run-

MicrowavesAntenna

Susceptor

Ceramic

Insulation

Fig. 27. Cylindrical microwave cavity with a ceramic sample and suscep-tors.

Fig. 28. (a) Temperature distribution for 1 kW microwave power. (b)Maximum and average temperature in the (coarse-grained) ceramic showthe evolution of a runaway instability.

away instability to occur are satisfied here. In practice thismeans that the process fails to yield an acceptable sinteredproduct.

On the other hand, the instability can be avoided, iffine-grained alumina with grain size 0.1�m is sintered with500 W microwave power. AsFig. 29shows, the temperaturedistribution in the fine-grained ceramic is very homoge-neous. This leads to a density near 100% and a fine grainsize below 0.14�m throughout the specimen. The reason forthe favorable behavior of the fine-grained material is that itstarts to sinter earlier. Since the heat conductivity increasessharply during the initial stages of densification, the heat isspread more effectively and hence no localization instabil-ity develops. A model which does not couple sintering withthe electromagnetic and thermal fields would not be ableto predict the difference between coarse- and fine-grained


Fig. 29. (a) Temperature distribution for 500 W of microwave power. (b)Maximum and average temperature in the (fine-grained) ceramic.

materials. The development of stable microwave sinteringprocesses for functionally graded materials without localover-heating can be substantially facilitated by the use ofsimulation tools like the present one.

5.3. Modeling of drying processes

After wet forming processes it is necessary to dry thegreen powder compact, before it can be processed further.Functionally graded materials are especially susceptible towarpage during drying. The capillary pressure of the liquidexerts an increasing pressure on the solid skeleton as the liq-uid content decreases, which causes the well-known shrink-age during drying. Since the rate of drying and the capillarypressure depend on the grain size and the green density, theshrinkage is usually inhomogeneous in a graded component,which leads to distortions and, in some cases, also to crackformation.

In the priority program a basis for modeling the warpageduring drying was developed. The aim was to be able tomodel the drying and deformation behavior of graded com-

ponents with a commercial finite-element program. A com-plete description of drying and the concomitant deforma-tions involves three coupled fields, namely the local watercontent, the temperature and the mechanical deformation.Various aspects of drying theory are described in[139,140].In a first step, a one-dimensional PC program was devel-oped. The program simultaneously solves the heat conduc-tion equation and Darcy’s equations for liquid and vaportransport in the porous solid. The thermodynamic propertiesof water (vapor pressure over curved surfaces, latent heat) aretaken into account. It turned out that under all practical dry-ing conditions the temperature was nearly uniform for a wallthickness of a few centimeters. Hence it is not necessary tosolve the heat conduction equation. It suffices to consider theglobal energy balance between the heat flowing into the bodythrough its surface, the latent heat of the evaporating liquidand the internal energy in the body. Hence in a numericaltime-stepping algorithm the (uniform) temperature changecan be calculated in each step directly from the current stateof the body and from the temperature and humidity of thesurrounding atmosphere. This greatly simplifies the analysis,since now only two fields (water content and deformationfield) remain which must be treated in a coupled manner.This two-field problem is, in principle, accessible to severalfinite-element codes. Nevertheless, the problem was furtherdecomposed into two one-field problems, namely the dryingsimulation itself and the deformation problem. The decou-pling is based on the assumption that the hydrostatic partof the stress is small compared to the capillary stress of theliquid. This excludes the description of situations in whichthe water is pressed out of the porous solid by applied forceslike in pressure filtration. On the other hand, the assump-tion is usually justified for industrial drying processes. Thedrying problem is then analogous to a heat conduction prob-lem, and the deformation problem is subsequently solvedas an elastic–plastic problem with superimposed volumetricstrains and plastic properties that depend on the local watercontent.

0 100 200 300 400 500 600 7000

5

10

15

20

25

Wat

er c

onte

nt in

wt-

%

Time in min

0

1

2

3

Shrinkage in %

Fig. 30. Water content and shrinkage during drying. Measured values andadjusted theoretical curves.


10

12

14

16

18

20

22

24

26

28

30

0 5000 10000 15000 20000 25000 30000 35000 40000

time in s

tem

pera

ture

in º

C

Fig. 31. Evolution of the temperature in the plate during drying.

The parameters of the model were determined froma set of experiments, in which the water content, theshrinkage and the temperature are measured for differenttemperatures and humidities of the atmosphere (Fig. 30).Further, stress–strain curves were measured in compres-sion tests for different water contents to determine theelastic–plastic properties of the wet powder body. Afterthe determination of the model parameters the drying andwarpage of homogeneous and functionally graded ma-terials can be described. As an example the drying of

Fig. 33. Warpage after drying. Initial geometry in the background.

Table 2Overview of processing methods for FGMs

Process Variability oftransition function

Layerthicknessa

Versatility inphase content

Type of FGM Versatility incomponent geometry

Powder stacking Very good M, L Very good Bulk ModerateSheet lamination Very good T, Mb Very good Bulk ModerateWet powder spraying Very good UT, Tb Very good Bulkc ModerateSlurry dipping Very good UT, Tb Very good Coating GoodJet solidification Very good M, L Very good Bulk Very goodSedimentation/centrifuging Good C Very good Bulk PoorFiltration/slip casting Very good C Very good Bulkc GoodLaser cladding Very good M Very good Bulk, coating Very goodThermal spraying Very good T Very good Coating, bulk GoodDiffusion Moderate C Very good Joint, coating GoodDirected solidification Moderate C Moderate Bulk PoorElectrochemical gradation Moderate C Good Bulkc GoodFoaming of polymers Moderate C Good Bulk GoodPVD, CVD Very good C Very good Coating ModerateGMFC process Very good M, L, C Moderate Bulk Good

a L: large (>1 mm); M: medium (100–1000�m); T: thin (10–100�m), UT: very thin (<10�m); C: continuous.b Depending on available powder size.c Maximum thickness is limited.

Fig. 32. Distribution of the water content in the plate after 650 s (a) and6600 s (b). The numbers give the range in which the water content varies.Only a quarter of the plate is depicted.

a graded circular plate with 60 mm diameter and 5 mmthickness was modeled in an atmosphere with 25◦C and50% humidity. The material is given model parameterssimilar to those of a sanitary ware material, and the vol-umetric drying strain is assumed to increase linearly from4 to 8% from the top of the plate to the bottom, whilethe permeability for liquid water increases by a factor 3.


As Fig. 31 shows, the calculated temperature of the plateinitially decreases to 19◦C due to the evaporation of water,and returns to ambient temperature as the evaporation ratedecreases. After about 40,000 s the water content in theplate reaches 1.6%, i.e. it is in equilibrium with the atmo-sphere and evaporation comes to a standstill.Fig. 32showsthe distribution of the water in the plate at different times,and Fig. 33 shows the warpage at the end of the dryingcycle. In a SPP project, extreme warpage was also experi-mentally observed for direct drying of plates with a particlesize gradient, since the capillary stresses of the liquid infine-grained material are much larger than in coarse-grainedmaterial[141]. In practice, planar plates could be obtainedby supercritical drying, since this process does not involvea direct liquid to vapor transition and hence avoids capillaryforces.

6. Concluding remarks

Processing of FGMs in the laboratory scale has reacheda considerable level of maturity. A range of process-ing methods is available today for almost any materialcombination. Which of these processing methods is themost appropriate depends not only on the materials in-volved, but also on the type and extension of the gra-dient, and the geometry of the required component (anoverview is given inTable 2). The German priority pro-gram has contributed to the current state-of-the-art ofFGM processing, especially in freeform processing andmetallurgical processing of FGMs. Additionally knowngradation methods were applied to the processing ofpolymer-based FGMs. Despite of all these achievements,the authors believe that there will be new challenges inthe future when applications for FGMs will evolve. Theseinclude:

(i) adaptation of manufacturing processes to mass produc-tion and up-scaling,

(ii) repeatability of production processes and reliability ofthe produced FGMs,

(iii) cost-effectiveness of production processes,(iv) quality control.

Modeling of production processes may contribute to solvesome of these problems, and first steps in this direction havebeen undertaken in the German priority program.

Acknowledgements

Support of the priority program “Functionally GradedMaterials” by the Deutsche Forschungsgemeinschaft isgratefully acknowledged. The authors would like to thankthe participants of the priority program for contributingmany of the illustrations used in this review.

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