Centrifugal synthesis and processing of functionally graded materials

A method through which we can synthesize Functionally Graded Materials (FGM). Such materials are made so that their composition changes gradually from one point to another, such as in the example of gradient index (GRIN) optical components. A novel aspect of our method is the imposition of a centrifugal force during the combustion synthesis of composite materials for structural, optical, or electronic applications, with the result that the composition and the particle size of the metallic (or ceramic) component changes continuously and across the thickness of the product. We have prepared such FGM as ZrO.sub.2 +Ni, ZrO.sub.2 +Cu and Al.sub.2 O.sub.3 +Cu.

BACKGROUND OF THE INVENTION 
The imposition of a compositional gradient across a piece of material 
results, expectedly, in a concomitant gradient in the properties of this 
material. Such functionally graded materials, or FGM, have recently been 
the focus of intense investigations, primarily in Japan. Although the 
initial emphasis was on the synthesis and processing of thermal barrier 
material for space applications {T. Hirano, Second Symposium for 
Functionally Gradient Materials, Jul. 1, 1988, Tokyo, The FGM Research 
Society, Kino Zairyo 8:15 (1988)}, e.g., the National Space Plane and 
shuttle engines, subsequent investigations have focused on other areas in 
which the application of FGM provides novel and effective solutions to 
existing materials problems. These latter areas include the use of FGM in 
nuclear fusion and fast breeder reactors (as first wall composite 
materials) {M. Seki, ibid, Kino Zairyo 8:7 (1988); T. Igari, et al., 
Proceedings of the First International Symposium on FCM; M. Yamanouchi, et 
al. (eds.), p.11 (1990)}, in electronic and magnetic applications 
(electro-ceramics, sensors), in optical applications (high performance 
laser rods, optical disks), in chemical applications (membranes, 
catalysts), in biomedical materials (tooth implants, artificial bones), 
and in joining applications (ceramic engines, heat and corrosion 
resistance coatings) {M. Niino, Kino Zairyo 7:31 (1987); T. Kawai, et al., 
Proceedings of the First International Symposium on FGM. M. Yamanouchi, et 
al. (eds.) p. 191 (1990); M. Yuki, et al., ibid, p. 203; M. Chigasaki, et 
al., ibid, p. 269}. The use of FGM in heat applications is exemplified by 
the proposed utilization of a TiB.sub.2 /Cu FGM in a reusable rocket 
engine {T. Hirano, supra}. The ceramic side (TiB.sub.2) of this FGM is 
designed to withstand 1500 K while the metallic (Cu) side is designed for 
operation at 300 K The advantage of a functionally gradient TiB.sub.2 /Cu 
material relative to two layer (TiB.sub.2 +Cu) and three layer (TiB.sub.2 
+50% TiB.sub.2 +Cu) alternatives is demonstrated by the results shown in 
FIG. 1 {T. Hirano, supra}. In cases of a multi-layer system, tensile 
stresses are generated at the interfaces while, in contrast, the FGM 
material experiences basically a compressive stress throughout. The 
existence of tensile stresses at the interfaces in the multi-layer systems 
is the primary cause of their failure. 
To provide a continuous or semi-continuous compositional gradient in 
functionally graded materials, several synthesis methods have been 
utilized. These include chemical and physical vapor deposition (CVD and 
PVD) {S. Ikeno, Second Symposium for Functionally Gradient Materials, Jul. 
1, 1988, Tokyo, The FGM Research Society, Kino Zairyo 8:19 (1988); T. 
Kirai, FGM News, Tokyo, May 1988, p. 10; M. Sasaki, et al., Proceedings of 
the First International Symposium on FGM, M. Yamanouchi, et al., (eds.) p. 
83 (1990); K. Fritscher, et al., ibid, p. 91; S. Uema, et al., ibid, p. 
237}, thermal and plasma spray {T. Fukushima, Second Symposium for 
Functionally Gradient Materials, Jul. 1, 1988, Tokyo, The FGM Research 
Society, Kino Zairyo 8:31 (1988); H. Steffens, et al., Proceedings of the 
First International Symposium on FGM, M. Yamanouchi, et al. (eds.), p. 139 
(1990); T. Fukushima, et a., ibid, p. 145; N. Shimoda, et al., ibid, p. 
151}, powder metallurgy techniques {N. Tsutsumi, ibid, Kino Zairyo 8:39 
(1988); R. Watanabe, FGM News, Tokyo, May 1988, p. 14; B. Ilschner, 
Proceedings of the First International Symposium on FGM, M. Yamanouchi, et 
al. (eds.), p. 101 (1990); R. Watanabe, et al., ibid, p. 107}, and 
self-propagating exothermic reactions {N. Sata, Second Symposium for 
Functionally Gradient Materials, Jul. 1, 1988, Tokyo, The FGM Research 
Society, Kino Zairyo 8:35 (1988); Y. Matsuzaki, et al., Proceedings of the 
First US-Japan Workshop on Combustion Synthesis, Jan. 11-12, 1990, Tsukuba 
Science City, Y. Kaieda, et al. (eds.), National Research Institute for 
Metals, Tokyo, p. 89; N. Sata, et al., ibid, p. 139; Y. Miyamoto, et al., 
ibid, p. 173; N. Sata, et al., Combustion and Plasma Synthesis of High 
Temperature Materials, Z. A. Munir et al., (eds.), p. 195. VCH Publishers, 
NY (1990); Z. Y. Fu, et al., Proceedings of the First International 
Symposium on FGM, M. Yamanouchi, et al. (eds.), p. 175 (1990); N. 
Yanagisawa, et al., ibid, p. 179}. Compared to others, the method of 
self-propagating exothermic reactions has the general advantages of 
simplicity, low cost, and the relative ease of preparing larger items. In 
this method, layers of reactants with gradually changing compositions are 
pressed together and then ignited at one end of the multi-layer ensembles 
to initiate a self-sustaining reaction front. The product then comprises 
regions in which the composition is constant but is incrementally 
different from that in the two adjacent layers. Thus, the composition 
changes in a step fashion from one end of the sample to the other. 
Despite its attractive features, this method suffers from two general 
disadvantages. The use of reactant layers with a successively changing 
composition can lead to the existence of discontinuities in composition at 
the interfaces. The second disadvantage relates to the nature of the 
self-propagating reaction: a compositional limit exists at which the 
reaction enthalpy is not sufficiently high to sustain the combustion wave 
{Z. A. Munir, American Ceramic Society Bulletin, 67:342 (1988); Z. A. 
Munir, et al., Materials Science Reports, 3:277 (1989); N. Sata, et al., 
Proceedings of the Fifth Symposium on High Temperature Materials 
Chemistry, Oct. 14, 1990, Seattle, Wash., W. Johnson, et al. (eds.), The 
Electrochemical Society}. This implies in the case of the TiB.sub.2 -Cu 
system, for example, that a gradient from pure TiB.sub.2 to pure Cu cannot 
be established by combustion synthesis. At higher copper contents, the 
reaction Ti+2B+Cu is not self-sustaining. To mitigate the first problem, 
thinner layers with small compositional differences can be used to prepare 
the FGM, but this solution adds complexity to the process and diminishes 
one of its attractive features. The second problem, on the other hand, 
does not have a simple solution. 
The prospect of inducing deliberately designed compositional gradients in 
optical and electronic materials is extremely attractive in the synthesis 
of special materials. For instance, gradient index (GRIN) optical elements 
are now a well accepted part of modern photonic and communication devices. 
These materials have a well controlled and continuous change in the 
refractive index and find applications in fiber optic couplers, 
photocopiers, miniaturized optical systems, and medical endoscopes. The 
effectiveness of such GRIN elements is strongly determined by the extent 
of the radial or axial refractive index change (.DELTA.n) that can be 
obtained in bulk disks or cylindrical preforms of the optical component. 
GRIN lenses with large refractive index variations (.DELTA.n&gt;0.1) and low 
dispersion are sought as optical blanks for processing of components with 
a variety of profiles and symmetry. 
In conventional ion exchange processes {S. N. Houde-Walter, et al., Applied 
Optics 25:3373 (1986)}, a glass rod of homogeneous composition is treated 
in a salt bath (NaNO.sub.3 +NaCl) to allow Na.sup.+ exchange at the 
surface. Thus, a surface layer will have a slightly different composition 
relative to the bulk and this in turn alters the refractive index. The 
range of .DELTA.n values obtained in this process is of the order of 
.apprxeq.0.01 to 0.05. For example, TiO.sub.2 -SiO.sub.2 rods, 2 to 3.5 mm 
in diameter, have been used to obtain lenses with .DELTA.n=0.015 to 0.025. 
In addition to CVD and sol-gel processing {M. Pickering, et al., ibid, 
25:3364 (1986); T. Edahiro, et al., U.S. Pat. No. 4,528,010, Jul. 9, 
1985}, other methods for the preparation of materials with a significant 
.DELTA.n include the co-melting of layers of powdered glasses of different 
composition to synthesize bulk optical quality materials with .DELTA.n=0.3 
to 0.5 {R. Blankenbecler, et al., Journal of Non-Crystal Solids, 129:109 
(1991)}. The composition profiles are not very smooth, however, and the 
process is completely dependent on adequate precision during the 
mechanical layering of the powders and melt interdiffusion under normal 
gravity conditions. 
Centrifugally-assisted FGM processing is a new synthesis technique for the 
preparation of these GRIN optical elements, especially since it permits 
layering of the constituents of the powdered batch under a strong gravity 
field (50 g.sub.o). The resulting GRIN material is continuously graded in 
composition and refractive index and the possibility of creating large 
.DELTA.n in bulk samples, through the innovative use of centrifugally 
assisted FGM processing, is a major advance over current synthesis methods 
for GRIN optical materials. 
Another important area in which FGM processing is desirable is quantum dot 
materials. In quantum dot materials, the band gap of a bulk semiconductor 
is shifted significantly by reducing its particle size to a value smaller 
than the exciton (electron-hole pair) Bohr radius. The most exciting 
possibility here is the potential for tailoring quantum devices by 
size-selection of the semiconductor in nanometer dimensions. Bulk quantum 
dot materials are composites of semiconductor particles (e.g., CdSe, GaAs) 
suspended in a ceramic or glass medium. The quality of quantum 
confinement, i.e., the optical density, is related to the concentration of 
semiconductor particles and it is here that FGM processing can result in 
some unique materials with a gradient in semiconductor quantum dot 
concentration. FGM processing also has the potential for creating 
quantum-confined structures with predictable quantum dot concentrations in 
various regions of the same bulk sample. The study of the optical 
absorption spectra of such samples can be expected to give useful insights 
into the band structure transitions in these materials. It will be 
particularly interesting to investigate the non-linear optical response in 
a bulk solid processed by FGM as the non-linearity can be investigated 
point to point by moving the laser beam across areas of variable optical 
density. 
A central feature of the method of the present invention is the use of a 
centrifugal force (F.sub.c) to prepare graded materials with controlled, 
continuous compositional gradients. The use of a centrifugal force in 
self-propagating synthesis has been shown to modify the process itself as 
well as the macroscopic characteristics of the product. However, the 
primary focus of the use of a centrifugal force has been, thus far, the 
separation of the product phases to obtain cast materials {A. G. 
Merzhanov, et al., Nauchnye Osnovy Materialovedniia, Moscow, p. 193 (1981 
)}. For the centrifugal force to have an effect, a product or an 
intermediate phase should be in the liquid state and the phases to be 
separated should differ in their specific gravity. An example of such a 
system is the thermite reaction: 
EQU Fe.sub.2 O.sub.3 +2Al=Al.sub.2 O.sub.3 +2Fe (1) 
The adiabatic temperature (3400 K) for this reaction exceeds the melting 
points of both product phases. However, under actual experimental 
conditions (where adiabatic conditions are not maintained) or with the 
addition of appropriate amounts of a diluent (in this case, Al.sub.2 
O.sub.3), the combustion conditions give rise to the formation of solid 
Al.sub.2 O.sub.3 and liquid Fe. Under such conditions, the application of 
a centrifugal force, F.sub.c, leads to the separation of the lighter phase 
Al.sub.2 O.sub.3 (.rho.=3.5 g. cm.sup.-3) to the top of the heavier phase, 
Fe (.rho.=7.86 g. cm.sup.-3). This process has been commercially utilized 
in Japan to place a corrosion-resistant coating (Al.sub.2 O.sub.3) on the 
inside of steel pipes {O. Odawara, Japanese Patent # JP55-341416 (1980); 
O. Odawara, Combustion and Plasma Synthesis of High Temperature Materials, 
Z. A. Munir, et al. (eds.), p. 179, VCH Publishers, NY (1990)}. 
Several investigations on the effect of a centrifugal force on the process 
of self-propagating synthesis have been made {A. G. Merzhanov, et al., 
Proceedings of the First US-Japan Workshop on Combustion Synthesis, Jan. 
11-12, 1990, Tsukuba Science City, Y. Kaieda, et al. (eds.), National 
Research Institute for Metals, Tokyo, p. 1}. However, the emphasis of 
these experimental studies has been the determination of the effect of 
F.sub.c on the parameters of the combustion process. No systematic study 
involving theoretical fundamentals of phase separation appears to have 
been made and no attempt has been made to use a centrifugal force to 
prepare functionally graded materials with controlled compositional 
distributions. 
An example of the effect of F.sub.c on the velocity of the combustion wave 
of the reaction described by Eq(1) is shown in FIG. 2 {B. B. Serkov, et 
al., Fiz. Gor. Vzryva, 4:600 (1968)}. For this reaction, the wave velocity 
increases almost linearly with increasing F.sub.c (in this figure, F.sub.c 
is represented by the ratio of the acceleration, g, to that of gravity, 
g.sub.o). This effect, however, is not universal and is a function of two 
factors with opposite effects: (a) an increase in rate due to 
gravity-enhanced permeation of the liquid phase ahead of the wave, and (b) 
an increase in heat loss due to increased convection in the liquid phases 
in the combustion zone. The net influence of F.sub.c depends on the 
relative contribution of these two factors. In highly exothermic reactions 
the velocity, u, increases dramatically with increasing F.sub.c (curve a), 
and in moderately exothermic reactions the influence of F.sub.c on u shows 
a maximum (curves b and c), as seen in FIG. 3 {S. A. Karataskov, et al., 
ibid, 6:41 (1985)}. The maximum is a consequence of the change in 
dominance of the two factors discussed above. At lower values of F.sub.c 
the process is dominated by liquid permeation and at higher values of 
F.sub.c it is dominated by heat loss due to convection. The importance of 
a gravitational force on combustion synthesis reactions is underscored by 
observations that for weakly exothermic reactions, a self-propagating wave 
can only be established under the influence of F.sub.c (curve d in FIG. 3) 
and for moderately exothermic reactions (those exhibiting a maximum in u 
vs F.sub.c) increasing the gravitational force can lead to the extinction 
of the combustion wave (curve b). 
Phase separation as a result of the application of F.sub.c depends on the 
strength of the force F.sub.c, the viscosity of the liquid phase, the size 
of the solid particles being separated, the difference between the 
densities of the liquid and solid phases, and the time during which the 
centrifugal force is in effect. Since the influence of this force is only 
seen when a liquid phase is present, the parameter of time, therefore, is 
dictated by the nature of the combustion process. FIG. 4 shows a schematic 
representation of the temperature profiles of two combustion reactions in 
which the effective times are indicated as t.sub.1 and t.sub.2 for the 
highly exothermic reaction 1 and the less exothermic reaction 2, 
respectively. In general, phases in systems with higher combustion 
temperatures can be separated easier than those with lower combustion 
temperatures. Since the combustion temperature can be controlled by 
dilution {J. B. Holt, et al., Journal of Materials Science, 21:251 
(1986)}, the effective time of the application of F.sub.c can be 
experimentally varied. 
Experimental observations on the effect of F.sub.c on phase separation are 
shown in FIG. 5 {Merzhanov, et al., supra). It is significant to note that 
the transition between "no separation" and total separation depends on the 
system investigated and can take place over a relatively narrow range of 
F.sub.c values (for the reaction WO.sub.3 +CoO+Al+C) or over a wide range 
of F.sub.c (for the reaction WO.sub.3 +Al+C). 
In a recent investigation {J. B. Hurst, NASA Technical Memorandum 102004, 
May 1989}, the effect of gravity on phase separation in products of 
combustion synthesis of nickel aluminides was examined. Cylindrical 
samples were ignited, in one case at the bottom and the other case at the 
top. Although no phase separation was detected (under 1 g.sub.o gravity), 
an interesting observation was made with regard to pore formation. While 
the total porosity of both types of samples was the same, those that were 
ignited from the bottom had markedly larger pores when compared to those 
samples ignited at the top. Qualitatively, these observations suggest that 
pore segregation occurs in the combustion (liquid) zone. In the case where 
the wave is propagating in an upward direction, pores that segregate to 
the top of the combustion zone are swept in the same direction as the wave 
and coalescence with other pores is likely. With a downward moving wave, 
pores that segregate to the top of the combustion zone are left behind 
with little likelihood of growth by coalescence. The effect of a 
gravitational force of higher than 1 g.sub.o on the size and distribution 
of pores in this or other systems was not investigated. 
SUMMARY OF THE INVENTION 
Functionally Graded Materials have been prepared by a variety of methods. 
None of these have used a gravitational force to effect the desired 
compositional gradient. Another novel aspect of our approach is to couple 
a gravitational force to the process of combustion synthesis. As long as 
the phases in the product differ in their specific gravity, an effective 
gradual distribution can be achieved under the gravitational force. 
Although FGM have been prepared by combustion synthesis, the approach 
taken thus far does not result in a smooth compositional gradient. 
The functionally graded materials, i.e., composites with a spatial gradient 
in compositions, of particular interest are metal-ceramic composites 
having a gradient in the density of ceramic particles embedded in the 
metal matrix. Such materials are heat resistant in the region where the 
ceramic concentration is high, and have high strength where the metal 
concentration is high. We are disclosing two novel techniques to produce 
FGMs by centrifuging particles into molten metal: (1) combustion synthesis 
with centrifugation, and (2) thermal melting of metals with centrifugation 
of particles into the melt. Preliminary results indicate the potential of 
these techniques to achieve smoothly-varying gradients of composition. 
Calculations based on mathematical models of phase separation (particle 
sedimentation) in molten metals are consistent with the experimental 
observations. 
To provide the details of one version of our invention, we use the example 
of preparing a functionally graded material/ceramic composite such as 
ZrO.sub.2 +Cu. For this we start by mixing copper oxide, CuO, and 
zirconium metal, Zr. The powders are then pressed together to form a 
cylindrically-shaped sample. The sample is then placed in a centrifuge 
capable of providing an acceleration of 100 g or more. Once the 
gravitational force is applied, the sample is ignited at one end and a 
combustion wave is established. In the system we have chosen, and in many 
other systems, at least one component of the reaction is in the liquid 
state. In this case the metal copper will be liquid. Because of the 
imposed gravitational force and the fact that the specific gravities of 
the two product phases (copper and zirconium oxide) are different, a phase 
separation will take place. The nature of the compositional gradient, 
i.e., its steepness, depends on the materials' properties (e.g., their 
specific gravities), on the combustion process (e.g., the combustion 
temperature), and on the magnitude and duration of the imposed 
gravitational field. In contrast to the method of formation of FGM by 
combustion synthesis, our method can produce graded materials in which the 
phases can change from pure ceramic to pure metallic, or in an all oxide 
system, gradients affecting important optical properties (e.g., refractive 
index) can be deliberately designed.

DETAILED DESCRIPTION OF THE INVENTION 
An understanding of the separation of particles of different sizes that are 
being formed and growing in a melt is needed for process design and 
optimization of centrifugal FGM processing. A mathematical treatment of 
particle phase separation (sedimentation) and the resulting compositional 
gradient is based on a distribution function for the particle size, f (r, 
z, t), which depends upon time, t, and the position, z, in the vessel, 
i.e., f (r, z, t) dr is the number of particles (per unit vol) in the size 
range (r, r, +dr) and located at (z, t). It is assumed that the particles 
are spheres of radius, r, and that z is measured in the direction of 
particle motion. The governing partial differential equation for such a 
particle-size distribution function is the same equation used in 
population-balance models {D. M. Himmelbau, et al., Process Analysis and 
Simulation-Deterministic Systems, Chapter 4, Wiley, NY (1 968)}, i.e., 
##EQU1## 
where .OMEGA.(r, z, t) is the rate of growth of particles, and G(r, z, t) 
is the net generation rate of the spherical particles. The terminal 
velocity of the spheres in the melt is written as 
##EQU2## 
in terms of the densities of the solid and liquid phases, .rho..sub.S and 
.rho..sub.L, the melt viscosity, .mu., and the centrifugal acceleration, 
g=ag.sub.o, expressed as a multiple of the gravitational acceleration, 
g.sub.o. The expression for v is based on the assumption that the 
particles are sufficiently far apart that particle--particle interaction 
can be neglected. Particle interaction (hindered settling) can be included 
with an appropriate functional form of the velocity, v, or with Kynch 
sedimentation theory {F. M. Tiller, AlChE Journal, 27:823 (1981)}. Other 
complications, such as convective effects for mixtures of buoyant and 
heavy particles and vessel wall influence, can also be treated 
mathematically {Y. T. Shih, et al., Powder Technology. 50:201 (1987)}. 
The particle-size distribution function is related to observable 
quantities. The volume fraction of particles is defined as 
##EQU3## 
and the volume flux of particles is given by 
##EQU4## 
in terms of the particle velocity v=.phi.r.sup.2. The particles are 
deposited when they reach the end of the vessel at z=L. The volume of 
deposit (per unit area) at time, t, can be written as 
##EQU5## 
where the deposit depth, h, grows with time, and is defined in terms of 
the deposit volume fraction, F.sub.s, 
##EQU6## 
For dense deposits, h&lt;&lt;L, and J (z=L, t) is a suitable approximation in 
Eq(6). 
The population balance equation in general requires a numerical solution, 
but some special cases of interest in centrifugal FGM processing can be 
solved analytically in closed form. In principle the predicted volume 
fraction gradient can be used to estimate other property gradients in the 
FGM, e.g., density, heat capacity, thermal conductivity, thermal expansion 
coefficient, refractive index, and elasticity constants. 
Results for the gradient in volume fraction of particles in the melt are 
illustrated by calculations based on the model system of Al.sub.2 O.sub.3 
particles in molten Al. The properties of the system are provided in Table 
I. 
TABLE I 
______________________________________ 
Properties of the Centrifugal Sedimentation of Al.sub.2 O.sub.3 
Particles in Molten Al. 
______________________________________ 
density.sup..dagger. of Al.sub.2 O.sub.3 particles 
.rho..sub.s 
= 3.97 g/cm.sup.3 
density.sup..dagger..dagger. of molten Al .rho..sub.L = 2.70 g/cm.sup.3 
viscosity.sup..dagger..dagger..dagger. of molten Al .mu. = 2.0 cp 
centrifugal acceleration g = 5 g.sub.o, 
g.sub.o = 980 cm/s.sup.2 
v = .phi.r.sup.2, with .phi. = 70550/s cm 
largest particle radius r.sub.o = 0.0001 cm 
length of vessel L = 1.00 cm 
final depth of deposit h.sub..infin. = 0.10 cm 
______________________________________ 
.sup..dagger. R. C. Weist, Handbook of Chemistry and Physics, 49.sup.th 
Ed., Chemical Rubber Co., p. B173 (1968). 
.sup..dagger..dagger. Idem, ibid, p. B172. 
.sup..dagger..dagger..dagger. G. H. Geiger, D. R. Poirier, Transport 
Phenomena in Metallurgy, AddisonWesley, p. 18 (1973). 
FIG. 6 shows the reduced volume fraction versus length coordinate, z, for 
an initial uniform and constant distribution of Al.sub.2 O.sub.3 particles 
of size less than r.sub.0 =1 .mu.m. For these rather small particles, the 
centrifugal phase separation at 5 g.sub.0 requires on the order of an hour 
for most particles to deposit. For shorter times, a well characterized 
gradient in the concentration of these particles can be established. The 
deposit layer grows with time as indicated at the right-hand side of FIG. 
6. Since the larger particles are deposited before smaller particles, one 
expects a gradient in particle size within the deposit. A gradient in 
particle size is also expected in the distribution prior to the complete 
phase separation. 
Thus, for a multisized particle sample, the FGM comprises a gradient in 
particle size as well as in particle volume fraction. Larger particles 
settle farther and faster into the melt. The presence of simultaneous 
gradients in volume fraction and particle size may have advantages for 
certain applications. For the case when particles within a size range are 
being generated during the centrifugal phase separation, the volume 
fraction gradient can be designed to contain particles of different sizes 
over the length of the sample (Case 3). 
The formulation of FGMs by centrifugal phase separation is demonstrated by 
three generic experiments: (a) particles generated throughout the sample 
by solid combustion caused by uniform heating (thermal explosion method), 
(b) inert particles settling into a melt, and (c) particles generated 
within a traveling combustion front ignited at one end of the sample. 
Several parameters serve as controlling factors for tailoring the 
properties of the FGM formed by centrifugal phase separation: centrifugal 
acceleration, particle size and size distribution, particle shape, 
particle density, melt density, melt viscosity (temperature), 
centrifugation time, and length of centrifugation chamber. 
In addition to the combustion synthesis methods for preparing FGMs, several 
nonreactive systems are suggested for embedding inert particles as a 
gradient in a metal matrix. The basic concept is simple: allow particles 
to be centrifugally driven through molten metal until the desired gradient 
is achieved, then cool and solidify the metal. The following cases are 
examined: 
1. Metal and ceramic particles are mixed together and placed in a 
phase-separation chamber. The temperature is raised above the metal 
melting point, and centrifugation causes movement of the particles through 
the molten metal. 
2. In the chamber (centrifuge tube), a layer of ceramic particles is placed 
over a layer of solid metal (or metal particles). The temperature of the 
vessel is raised above the metal melting point, and centrifugal force 
causes the ceramic particles to settle into the molten metal. The 
direction of particle motion is determined by the relative densities of 
the particles and the melt. The different initial conditions of 1 and 2 
will in general lead to different gradients. 
3. The size and shape of sedimented particles can influence and control the 
properties of the resulting FGM. Typically, smaller particles will provide 
smoother gradients and yield more desirable properties. For additional 
strength, fibrous particles can be sedimented into the metal. For certain 
applications, it may be desirable to sediment ceramic powder to one end of 
a sample and fibrous particles to the other end. The densities of the 
particles relative to the melt determine the direction of sedimentation. 
These three cases are presented to illustrate key features of the 
mathematical model for the phase-separation by centrifugation: (1) 
particles are initially dispersed in the melt, (2) a layer of particles is 
introduced at the entrance of the phase-separation chamber, and (3) 
particles are generated uniformly in the melt. The distribution of 
particle sizes, for simplicity in explaining basic concepts, is a 
rectangular distribution (equal number of particles of different sizes up 
to radius r.sub.o). 
Case 1 
Nongrowing particles (.OMEGA.=0) initially are distributed in the 
phase-separation chamber according to 
EQU .function.(r, z, t=0)=.function..sub.o (r, z) (8) 
and no particles are generated (G=0). The solution to Eq(9) is: 
EQU .function.(r, z, t)=.function..sub.o (r, z) [1-u(t-z/.nu.)](9) 
where the step function is u(x)=1 if x&gt;0 and 0 if x&lt;0. If the initial 
particle-size distribution is rectangular and uniform (independent of z), 
i.e., 
EQU .function..sub.o (r, z)=.function..sub.c for 0.ltoreq.r.ltoreq.r o and 0 
for r&gt;o (10) 
then the step function in Eq(9, since it depends on v=.phi.r.sup.2, can be 
replaced with u(r-r'), where r'&lt;r.sub.o is defined by 
##EQU7## 
The interval of integration in Eqs (4) and (5) is thus (r', r.sub.o). 
Performing the integrations and normalizing the volume fraction gives 
##EQU8## 
The deposit density is defined in terms of the depth at infinite time, 
h.sub..infin., i.e., F.sub.S =L F(t=0)/h.sub..infin. ; one obtains 
##EQU9## 
FIG. 6 shows the volume fraction (composition) gradient and the deposit 
depth at various times. The volume fraction of particles increases along 
the length coordinate in proportion to z.sup.2. The properties of Al.sub.2 
O.sub.3 particles in molten Al for the centrifugal phase separation are 
presented in Table I. 
Case 2 
If a layer of particles is introduced at z=0 the governing equation for 
.OMEGA.=0 and G=0 is 
##EQU10## 
with the initial condition, f(r, z, t=0)=0. The boundary condition 
EQU .function.(r, z=0), t)=.function..sub.o (r) [u(t)-u(t-t.sub.o)](15) 
describes a pulse of particles of duration t.sub.o that enters the 
phase-separation chamber. The solution can be found by Laplace 
transformation 
EQU (r, z, t)=.function..sub.o (r)[u(t-z/.nu.)-u(t-t.sub.o -z/.nu.)](16) 
and the step functions place limits on the values of r for integrations 
over r. The volume fraction becomes 
##EQU11## 
with F.sub.o =(.sigma./3)f.sub.c r.sub.o.sup.4. The sediment depth can be 
determined by the same method used for Case 1. The gradient and sediment 
depth at various times are shown in FIG. 7. At a very early time, the 
gradient decreases, while at long times the gradient increases in 
proportion to z.sup.2 as the particles make their way to the end of the 
vessel, where they are deposited. 
Case 3 
If nongrowing particles are being generated at a constant rate, the 
governing equation is 
##EQU12## 
with the initial condition f(r, z, t=0)=0, and the boundary condition f(r, 
z=0, t)=0. The solution is readily found when the generation term is 
uniform in space, i.e., G(r, t) 
##EQU13## 
For a constant generation term defined as G=G.sub.o for 
0.ltoreq.r.ltoreq.r.sub.o and 0 for r&gt;r.sub.o, the solution simplifies to 
EQU .function.(r, z, t)=G.sub.o [t-(t-z/.nu.)u(t-z/.nu.)] (20) 
Relative to F.sub.o =(2/3).pi.G.sub.o Lr.sub.o.sup.2 /.phi., the volume 
fraction is 
##EQU14## 
With the substitution of Eq(21) this expression yields the limit E=z/L as 
t becomes infinite, i.e., the generation rate is balanced by the 
sedimentation rate, and then the volume fraction increases linearly with 
the length coordinate. The volume fraction gradient and the sediment 
thickness are shown in FIG. 8. 
These mathematical models permit the properties of the final FGM to be 
tailored to specification. For any optimization experiment, the parameters 
that control the gradient of composition are, according to Eq(3), the 
centrifugal acceleration, g, the time of centrifugation, t, the chamber 
length, L, the particle size and shape distribution, the particle and 
liquid densities, and the liquid viscosity (dependent on temperature and 
composition of the melt). The mathematical model assists in suggesting how 
these parameters should be selected to optimize the FGM properties. 
Preferred systems for these nonreactive centrifugal FGM methods are based 
on Al (m.p.=660.degree. C.). Due to their high strength-to-weight ratio, 
aluminum matrix composites are used extensively in automotive and 
aerospace structural applications at ambient temperatures. A high priority 
in materials research is to develop aluminum composites for applications 
at above-ambient temperatures. The need exists to replace existing Ti 
alloys and stainless steel in engine components. Aluminum-matrix 
composites with gradient properties can combine the desirable features of 
strength, wear-resistance, temperature stability, and toughness. The 
method of the present invention is used to make aluminum-matrix FGMs with 
embedded gradients of either C, SiC, or Al.sub.2 O.sub.3. Whiskers or 
particles of these and other ceramics are readily available. For higher 
temperature applications titanium or nickel comprises the metal matrix. 
The calculations presented above are based on a simplified mathematical 
model, and are included for illustration purposes. This rudimentary 
picture of FGM manufacture by centrifugal phase separation provides 
qualitative insight into the key variables and fundamental processes. A 
realistic mathematical model to describe in detail the results of 
optimization experiments will likely be considerably more complex. The 
following extensions are suggested: 
a. The particle size distribution will be generalized to: 
i. a linear distribution f.sub.o (r)=f.sub.c (1-r.sub.o) for r&lt;r.sub.o and 
0 for r&gt;r.sub.o ; 
ii. an exponential distribution, f.sub.o (r)=f.sub.c exp (-r/r.sub.o); 
iii. a gaussian distribution, f.sub.o (r)=f.sub.c exp [-r-ro).sup.2 
/2.pi.]. 
b. The growth term will be generalized to: 
i. a linear distribution, G.sub.o (r)=G.sub.c (1-r.sub.o) for r&lt;r.sub.o and 
0 for r&gt;r.sub.o ; 
ii. an exponential distribution, G.sub.o (r)=G.sub.c exp [-r-r.sub.o), 
iii. a gaussian distribution, G.sub.o (r)=G.sub.c exp [-(r-ro).sup.2 
/2.sigma.]. 
c. The growth term will be incorporated, e.g., based on diffusion theory 
for particle growth, and expressions for .OMEGA. will be included in the 
population balance equation (Eq2). 
d. Embedding of rod-shaped particles (fibers) in the FGM will be 
considered. 
e. Particle interactions (hindered settling by Kynch theory) will be 
applied to the quantitative description of centrifugal production of FGMs. 
f. Simultaneous phase separation of light and heavy particles will be 
considered: particles heavier than the melt will move with the centrifugal 
force, and particles lighter than the melt will move opposite to the 
centrifugal force. 
Preliminary experiments whose aim was to demonstrate the feasibility of 
establishing a compositional gradient by the centrifuge-assisted 
combustion synthesis method have been conducted. A thermite reaction 
(2NiO+Zr=2Ni+ZrO.sub.2) was initiated in a thermal explosion method while 
the system was under the effect of a 50 g.sub.o centrifugal force. X-ray 
analysis of the product showed a clear separation of the phases (metal and 
oxide). The ratio of intensity of the peaks 
##EQU15## 
changed by more than a factor of two from one end of the sample to the 
her, with intermediate values in between. Other systems (e.g., CuO+Zr) 
were also examined with the results, again, showing clearly that a 
compositional gradient can be established under the action of a 
gravitational force. 
As shown above (Eq(3)), phase distribution can be examined through 
calculations of the distance of travel of a solid particle in a liquid 
phase, i.e., 
##EQU16## 
Direct control over the parameters g and .rho..sub.1 and .rho..sub.2 can 
be made and indirect control over t and .mu. can be attained through 
changes in the experimental conditions (higher combustion temperatures 
lead to higher t values and lower .mu. values). Control of the particle 
size is not possible in a direct way. However, as in the classical 
nucleation and growth phenomena, particle size depends on the degree of 
supercooling and is, therefore, expected to be influenced by the rate of 
cooling of the sample which in turn is affected by the rate of heat loss. 
The latter can be altered experimentally by changes in the thermal 
insulation of the sample. Experimental evidence has been provided showing 
that higher heat losses associated with convective phenomena under high 
F.sub.c results in smaller particle size of a solid carbide phase {A. G. 
Merzhanov, et al., supra}. Higher heat losses resulted in mixed carbide 
particles (TiC+Cr.sub.3 C.sub.2) of sizes in the range 3-4 .mu.m. When 
heat losses were not large, the particles were in the size range 10-20 
.mu.m. It should be emphasized that differences in heat loss in these two 
cases were a direct consequence of the application of a gravitational 
force. 
The effect of a density difference provides an interesting area of 
development, with the following system as an example: 
EQU CrO.sub.3 +(2+x)Al=Al.sub.2 O.sub.3 +CrAl.sub.x (23) 
The density of alloy product (CrAl.sub.x) depends on its composition (i.e., 
the value of x) and can be changed such that it can be greater or lower 
than that of Al.sub.2 O.sub.3 (.rho.=3.5 g. cm.sup.-3). It has been 
observed that with lower values of x, the oxide (Al.sub.2 O.sub.3) 
separates at the top and with higher values of x, Al.sub.2 O.sub.3 
separates to the bottom {A. G. Merzhanov, et al., Proceedings of the First 
US-Japan Workshop on Combustion Synthesis, Jan. 11-12, 1990, Tsukuba 
Science City, Y. Kaieda, et al. (eds.), National Research Institute for 
Metals, Tokyo, p. 1}. At intermediate compositions no phase separation has 
been observed. The effect of a centrifugal force on phase distribution in 
systems near this intermediate range is investigated. With a small 
difference between .rho..sub.1 and .rho..sub.2 the effect of the 
gravitational force can be clearly ascertained. 
The fundamental aspects of gravitational-force enhanced phase distribution 
were explored in another system, namely, 
EQU a(Ti+C)+b(xNi+yAl)=aTiC+bNi.sub.x Aly (24) 
The effect of the metallic content (ratio of b/a) and the composition of 
the metal phase (ratio of x/y) on the property of the product has already 
been examined {S. D. Dunmead, et al., Journal of Materials Science, 
26:2410-16 (1990)}. It was demonstrated (see FIG. 9) that the particle 
size of the ceramic phase (TiC) depended on the ratio b/a: higher ratios 
resulted in smaller particles. This system was investigated under a 
centrifugal force to ascertain the predicted effect of particle size on 
the phase distribution in the resulting functionally gradient material. 
The ratio b/a dictates primarily the combustion temperature but the ratio 
x/y influences the lowest temperature at which a liquid phase is possible. 
Thus, changes in both ratios can be used to alter the parameter of t and r 
in Eq(22). 
Different chemical systems can be selected to provide fundamental data on 
the synthesis of ceramic-ceramic FGM composites as distinct from 
ceramic-metal composites discussed above. Shown in Eq(25) is a modified 
thermite reaction to synthesize Mo.sub.2 C and Al.sub.2 O.sub.3 : 
EQU 2MoO.sub.3 +4Al+C=Mo.sub.2 C+2Al.sub.2 O.sub.3 (25) 
For this reaction, the calculated adiabatic temperature (4220.degree. C.) 
is above the boiling point of Al.sub.2 O.sub.3 (2980.degree. C.). However, 
by the addition of Al.sub.2 O.sub.3 as an inert diluent the combustion 
temperature may be reduced below 2980.degree. C. but still above the 
melting point of Mo.sub.2 C (2687.degree. C.). Therefore, over a certain 
temperature range it is possible to have two liquids. The final structure 
of the product is determined by the miscibility of these liquids at 
elevated temperatures. By adding larger amounts of Al.sub.2 O.sub.3 to the 
reactant mixture, the temperature can be reduced so that the products at 
the combustion temperature consist of solid Mo.sub.2 C and liquid Al.sub.2 
O.sub.3. The rate of nucleation and growth of the carbide particles is 
governed by the sequential reactions: (1) 2MoO.sub.3 
+4Al.fwdarw.2Mo+2Al.sub.2 O.sub.3 and (2) 2Mo+C.fwdarw.Mo.sub.2 C. The 
compositional gradient in the FGM is established by the same experimental 
parameters as discussed above. Because of the exothermicity of this 
reaction (Eq(25)), significant material loss is encountered through small 
explosions (spatter). The effect of the centrifugal force could modify the 
magnitude of this phenomenon. 
The work described above is carried out under two general combustion 
synthesis modes: thermal explosion and wave propagation {Munir (1988), 
supra}. Under the former condition, the entire sample is heated up to a 
temperature when the reaction occurs simultaneously throughout the sample. 
This condition provides for liquid formation, and hence phase separation, 
throughout the sample. Phase distribution, therefore, can occur across the 
entire sample as long as a liquid phase is present. The latter 
consideration (presence of a liquid phase) is governed by the combustion 
temperature, the nature of phase equilibria in the system under 
investigation, and by heat loss. 
Under the wave propagation mode, the samples are ignited at one end and a 
combustion front allowed to propagate. The effect of a centrifugal force 
on phase distribution for the case of a wave propagating in the same 
direction and in the opposite direction to the force of acceleration will 
differ. Under a wave propagation mode, the presence of a liquid phase 
would be primarily restricted to the wave front and thus phase separation 
is also restricted to that region. Therefore, in addition to the other 
parameters mentioned above, the velocity of wave propagation becomes a 
factor governing phase distribution. That phase distribution is possible 
under a wave propagation mode has been shown experimentally {A. G. 
Merzhanov, et al., supra}; however, the effect of the direction of the 
wave relative to the force of acceleration has not been investigated. It 
is expected that this effect will depend on the density of the (solid) 
particles relative to the density of the liquid phase. In this regard, the 
system described by Eq(23) offers the potential for controlling this 
effect by varying the value of x. 
Although so far chemical compositional gradient in FGM has been emphasized, 
this work can be extended to examine the effect of a centrifugal force on 
the distribution of porosity in combustion synthesized materials. As 
indicated above, there is experimental evidence that the size distribution 
of pores is influenced by a 1 g.sub.o gravitational field in 
nickel-aluminum alloys {Hurst, supra}. In principle, distribution of 
porosity is governed by the same theoretical criteria as those applicable 
to phase separation and the prospect is there that gradients of pores with 
given size and number distributions can be effected through the proposed 
research. Among other areas, materials with porosity gradients are of 
interest in biological (e.g., dental) implant applications {M. Kibrick, et 
al., Journal of Oral Implants, 6:172 (1975); Idem, ibid, 7:106 (1977)). 
The FGM processing of SiO.sub.2 -TiO.sub.2 compositions for use in 
electro-optic FGM systems was also investigated. It is well known that 
very low attenuation silica glasses with dopants (TiO.sub.2, P.sub.2 
O.sub.5, GeO.sub.2, etc.) can alter the refractive index of bulk glasses 
used for fiber optic communication systems. Ti O.sub.2 and Nb.sub.2 
O.sub.5 are especially effective in causing a steep increase in the 
refractive index, and TiO.sub.2 additions (at levels of up to 5 mole %) 
also decrease the thermal expansion coefficient of the glass (see FIG. 
10). Binary silica glasses (with TiO.sub.2, GeO.sub.2) and ternary systems 
have been developed for GRIN optics. Gel-derived leached glasses are 
usually sintered to dense components (typically rods 3 mm diameter by 10 
mm long) with .DELTA.n up to 0.1 (maximum). In the present research 
SiO.sub.2 -TiO.sub.2 powders are blended in various amounts and prepared 
for centrifugally-assisted FGM processing. Powders are charged into 
graphite molds and centrifugally accelerated to create multi-gravity 
(20-50 g.sub.o) environments. At the same time, the sample is heated to 
temperatures in excess of 1750.degree. C. in a chemical oven (heat 
supplied by a combustion reaction of, e.g., Ti+C) {U. Anselmi-Tamburini, 
et al., Journal of Applied Physics, 66:5039 (1989)}. This temperature 
exceeds the melting point of SiO.sub.2 (.about.1725.degree. C.) but not 
that of TiO.sub.2 (.about.1850.degree. C.). TiO.sub.2 particles suspended 
in the viscous SiO.sub.2 rich melt start settling due to the high g 
environment of the melt (.rho. for TiO.sub.2 =4.0 g. cm.sup.-3 and .rho. 
for SiO.sub.2 =2.2 g. cm.sup.3). These processing conditions thus create a 
melt with a gradient in the concentration of TiO.sub.2 varying from almost 
pure SiO.sub.2 at one end to 10 to 15 wt % TiO.sub.2 at the other. The 
solubility of TiO.sub.2 in liquid SiO.sub.2 at the processing temperature 
of 1750.degree. C. is limited to .apprxeq.18 wt % (to maintain a single 
phase liquid). The process of rapid TiO.sub.2 -SiO.sub.2 liquid formation 
is assisted by a eutectic in the phase diagram at .apprxeq.1550.degree. C. 
The settling and parallel reactions of TiO.sub.2 particles in the 
SiO.sub.2 rich melt occur quite rapidly in the FGM process and a 
composition profile develops in the melt due to the high g environment. 
Subsequent cooling of the melt gives a macroscopic glass blank with a 
gradient in refractive index because of the TiO.sub.2 concentration 
gradient. 
The FGM synthesized bulk SiO.sub.2 -TiO.sub.2 samples is analyzed for 
microchemical composition and optical properties. Electron beam microprobe 
analysis for composition is performed at 10 .mu.m intervals to obtain a 
composition profile of each sample. Parallel refractive index profiles are 
obtained by standard interferometric methods. Interference fringes across 
the sample provides point to point refractive index data. A key variable 
in the settling of TiO.sub.2 particles due to the FGM centrifuge 
processing is melt viscosity. Since viscosity has an exponential 
dependence on temperature, several test temperatures (in the range of 1750 
to 1850.degree. C.) are attempted to determine the effect of viscosity on 
composition gradient. 
Quantum dot electronic materials processing by FGM involves the same 
procedure but the system chosen is a special composition of a silicate 
glass mixed with CdSe particles. Following FGM processing in the 
centrifuge, samples are evaluated for optical absorption and non-linear 
response. The microstructure of these materials is also studied by TEM 
techniques using High Voltage Electron Microscopy (HVTEM). 
One of the advantages of FGM systems is that they provide a workable 
alternative to layered structures with regard to thermal stresses. It is, 
therefore, not surprising that considerable attention has been focused on 
this area. Investigations on thermal stress included the optimal design of 
FGM to reduce stress {T. Hirano, et al., Proceedings of the International 
Symposium on Space Technology and Science, Sapporo, Japan (1988)}, and the 
application of elasticity theory and finite element analysis to determine 
the effects of thermal mismatch in a composite {O. Kimura, et al., 
Proceedings of the First International Symposium on FGM, M. Yamanouchi, et 
al. (eds.), p. 359 (1990); M. Sasaki, et al., ibid, p. 83}. It has been 
shown that the thermal expansion coefficient of a TiC-SiC composite is 
nearly a linear function of composition {C. Kawai, et al., ibid, p. 77}. 
Different analytical techniques have been utilized to assess the extent and 
location of thermal stress in FGM {Kawai et al., supra}. These include the 
use of x-ray diffraction for surface stress determination and the use of 
SEM to observe thermally-generated cracks. The functionally graded 
material prepared in accordance with the present invention is 
characterized with regard to thermal stress by utilizing a well 
established technique {Sasaki, et al., supra}. Specimens are cooled with 
liquid nitrogen at one end and heated by an infrared lamp at the other 
through a cyclic mode. At different intervals the specimens are examined 
microscopically for crack formation. The results obtained from these 
investigations are compared to those obtained from another set of 
experiments in which FGM is made by combustion synthesis using the 
"layered" approach. 
EXAMPLE 1 
The following example shows the use of centrifugal force during combustion 
synthesis to obtain functionally-graded materials. Composites formed by 
the reaction 2Al+CuO=Al.sub.2 O.sub.3 +(3+x)Cu were synthesized in a 
centrifuge. The effects of diluent content, x, relative density of the 
reactants, and the particle size of CuO were investigated. Graded zones 
between the ceramic and metallic phases were obtained under a given set of 
these parameters. Phase separation times were calculated from 
sedimentation theory and discussed in light of experimental observations. 
I. EXPERIMENTAL MATERIALS AND METHODS 
Al.sub.2 O.sub.3 /Cu composites were synthesized via the thermite reaction, 
EQU 2Al+3CuO+xCu.fwdarw.Al.sub.2 O.sub.3 +(3+x)Cu (26) 
In Eq(26), x represents the amount of copper diluent added to the thermite 
reaction, and had values of 4, 6, 7 and 8. The aluminum powders used were 
99.5% pure and had a sieve designation of -325 mesh. Two powders of CuO 
were used. The first had a purity of 99.99% and a particle size range of 
43-841 .mu.m, while the second was 99+% pure with a particle size of &lt;5 
.mu.m. The copper (diluent) powders were 99% pure and had a sieve 
designation of -325 mesh. For the stoichiometric ratios indicated in 
Eq(26), powders were mixed and the samples were prepared in two ways. In 
one method the powder were pressed to form cylindrical pellets (1 cm in 
diameter and 1 cm high) with a relative density of 55%. The resulting 
pellets were placed in a graphic cylinder, which was subsequently placed 
inside a furnace. The second method of sample preparation involved pouring 
loose powder mixtures directly inside the graphite cylinder. These powders 
were either shaken to give a relative density of 37%, or were left as 
poured resulting in a relative density of about 20%. 
The graphite cylinder containing the sample in either form (pellet or loose 
powder), was placed inside a specially designed furnace. The sample 
assembly was covered with insulation and placed inside a large ceramic 
container equipped with an inlet and outlet for argon gas flow. The entire 
set-up was positioned inside the working arm of a centrifuge. The 
centrifuge used for this study has a one meter-long arm holding the 
"working budget". A second arm provided the counter-balance. The maximum 
centrifuge acceleration, g, is 100 g.sub.o, where g.sub.o is the 
gravitational acceleration. The entire sample enclosure was purged with 
argon gas before staring a centrifuge experiment. When the acceleration 
reaches the desired g level, the process of heating sample is initiated. 
The temperature was raised to the ignition point, which for Eq(26) is 
around 900.degree. C. Argon gas flow is maintained until the sample 
temperature has decreased to near room temperature. 
After being removed from the centrifuge, samples were examined and analyzed 
by optical microscopy, scanning electron microscopy (SEM), x-ray mapping, 
computerized image analysis, and x-ray diffraction analysis (XRD). 
II. RESULTS AND DISCUSSION 
When the reactants of Eq(26) were in the form of a compacted pellet (55% 
relative density), the product contained non-segregated Al.sub.2 O.sub.3 
and Cu phases dominated by an interconnected ceramic structure. However, 
when the reactants were in the form of a loose powder (37% relative 
density), the results were significantly different. For x=6 or 7, the 
product contained three regions. The bottom region was dense Cu, the top 
region was relatively, porous Al.sub.2 O.sub.3, and in between these there 
was a transition region showing a graded composition. FIG. 11 shows an SEM 
micrograph of the graded region between the ceramic and metallic phases 
for the reaction (Eq(26)) with x=6. Nearly spherical Al.sub.2 O.sub.3 
particles are distributed according to size, with the largest near the 
boundary between the transition region and the ceramic region. The 
transition region and the metallic region (Cu) are nearly fully dense 
while the ceramic region contained porosity, as can be seen in FIG. 11. A 
microprobe analysis of the compositional change between the upper and 
lower ends of the FGM zone (transition region) is shown in FIG. 12. The 
atomic concentration of Cu increases from zero at the upper interface of 
the FGM zone to nearly 100% at the lower interface. The aluminum 
concentration has an opposite trend, decreasing from 100% at the upper 
interface to nearly zero at the lower interface. 
A graded distribution of the ceramic particles for the case of x=7 was also 
obtained. However, a major difference between the two cases relates to the 
size of the alumina particles. The difference between the two systems can 
be better demonstrated by plots of Al.sub.2 O.sub.3 particle size 
distributions for the two cases. These are seen in FIGS. 13 and 14 for x=6 
and 7, respectively. The plotted results were obtained by a computerized 
image analysis where the average particle size was determined in layers 
sectioned successively within the FGM zone. In both cases, the particle 
size decreased by a factor of two from the ceramic side to the metallic 
side in the FGM zone. However, the Al.sub.2 O.sub.3 particles in the x=6 
system are significantly larger and their distribution varies inversely 
with distance cubed, as contracted to a nearly linear function for the 
case with x=7. The difference in particle size between the systems (x=6 
and x=7) is attributed to two related factors: combustion temperature and 
alumina content. For x=6, the adiabatic combustion temperature is 2694K 
and for x=7, the corresponding value is 2511K Obviously, the Al.sub.2 
O.sub.3 content is also lower for the case of x=7. Both of these factors 
favor the growth of larger alumina particles for the case of lower 
dilution (i.e. x=6). 
As was pointed out earlier, when compacted powders (55% relative density) 
were used, the product showed a network of inter-connected ceramic 
particles and no compositional gradient. With powders having a relative 
density of 37%, the product contained an FGM zone as described above. With 
the use of powders with a still higher porosity (relative density of about 
20%), the produce consisted of totally segregated ceramic and metallic 
phases. It is not clear whether these observations are related to the 
possible role of porosity in sweeping the ceramic phase to the top or they 
are the consequence of the larger ceramic interparticle distance in the 
product. At this point the role of porosity in the process of formation of 
FGM's through a centrifugally-assisted method is not clear. 
III. ANALYTICAL EVALUATION OF FGM FORMATION IN A CENTRIFUGAL FIELD 
The analysis of the Al.sub.2 O.sub.3 particle size distribution within a 
matrix of copper resulting from a centrifugally-assisted thermite reaction 
was made utilizing the population balance equation {D. M. Himmelblau, et 
al., "Process Analysis and Simulation-Deterministic System", Wiley, Chap. 
4 (1968)}. 
##EQU17## 
where f is the particle size distribution function, v is the particle 
terminal velocity, .OMEGA. is the growth rate of the particle, A is the 
net generation rate of particles, t is time, r is the particle radius, and 
z is distance. Application of Eq(27) is based on the following simplifying 
assumption: (a) the separation process begins immediately following the 
initiation of the reaction, (b) small-sized nuclei form in the homogeneous 
low density pellet first, then the sample shrinks to the high density 
product. Thus the nucleated particles are assumed to experience no growth 
during the densification stage because of its short duration. We consider 
the formation of larger particles to be the consequence of particle 
agglomeration. With these assumptions, a Laplace transformation of the 
population-balance model gives the relationship of the initial particle 
size distribution function, f.sub.o, to the final distribution function, 
as: 
EQU .function.(r, z, t)=.function..sub.o (r, z)[1-U(x)] (28) 
where x=t-z/v. The term U(x) is a step function with U(x)=for x&gt;o (t&gt;z/v) 
and U(x)=o for x&lt;o (t&lt;z/v). In Eq(28), v is the terminal velocity of the 
particle, t is time, and z is distance. Assuming f.sub.o to be in the form 
EQU .function..sub.o (r)=br.sup.a (29) 
the expressions for the volume fraction of particles in the sample, F.sub.o 
(z, t=o) and F(z, t) are: 
##EQU18## 
Normalization of the volume fraction (at any t) relative to the initial 
value gives 
##EQU19## 
where r.sub.M and r.sub.m are the maximum and minimum particle size, 
respectively, and r.sub.z is defined by 
##EQU20## 
where L is the length of the graded zone, t is time, and .phi. is defined 
by Stokes' law as 
##EQU21## 
The terminal particle velocity, V.sub.1 is in turn defined by 
##EQU22## 
where .rho..sub.S, .rho..sub.l, are the densities of the solid (Al.sub.2 
O.sub.3) and liquid (Cu) phases, respectively, g (=ag.sub.o) is the 
centrifugal acceleration, .mu. is the liquid phase viscosity, and r is the 
particle radius. 
The normalized volume fraction, E, can be calculated from experimental 
results. The calculation requires knowledge of F.sub.o (z, t) and F.sub.o 
(z, t=0) at any given z value. The former can be obtained from image 
analyses of sections of samples (i.e. at various z values). However, 
F.sub.o cannot be determined experimentally but an appropriate value can 
be calculated from the initial stoichiometry, Eq(26), assuming the product 
to be fully dense mixture of Al.sub.2 O.sub.3 and Cu. For x=6 and 7, 
F.sub.o is 28.74 and 26.63% by volume Al.sub.2 O.sub.3. Thus assuming the 
2-dimensional image analyses to represent volumetric distributions, E 
values are calculated as a function of z, as shown in FIGS. 15(a) and (b) 
for systems with x=6 and 7, respectively. 
Through a least-squares fit of the E values, two experimental parameters of 
the separation process can be calculated from Eq(32). These are the 
particle size exponent, a, and the separation time, t. The last parameter 
is implicit in the definition of r.sub.z in Eq(32). The calculated values 
for "a" and t for x=6 are -1.8 and 0.61s, respectively. The corresponding 
values for x=7 are 2.8 and 0.27s. 
The calculated times are the durations of the separation process of the two 
x values. The separation process, of course, takes place only when the 
copper is in the liquid phase, and thus the total time when the sample 
temperature is at or above 1083.degree. C. is important. Attempts to 
measure the temperature profile during the centrifuge experiments were not 
successful. Determinations of temperature profiles made at 1 g.sub.o and 
in a non-flowing argon atmosphere showed that the during when 
T.gtoreq.1083.degree. C. is 12 and 7s for the systems with x=6 and 7, 
respectively. These times are higher than the calculated separation times 
by factors of about 20 and 26, respectively. A complete phase separation 
would take place if the copper remained in the molten state for the times 
indicated by the 1 g.sub.o temperature profiles. However, a simple heat 
transfer analysis {W. Lai, MS thesis, University of California, Davis, 
Calif. (1996)} shows that in the presence of a flow of argon gas, 
convective heat loss could reduce the times by a factor of about 30 {R. B. 
Bird, et al., Transport Phenomena, Wiley, New York (1960)}. When taken 
into account, heat loss would reduce the length of the separation process 
to 0.4 and 0.21s for the cases of x=6 and 7, respectively. These 
approximately calculated values are in general agreement with those 
obtained from Eq(32).