Controlling Mechanical Properties of Porous Mullite/Alumina

Controlling Mechanical Properties of Porous Mullite/Alumina MixturesVia Precursor-Derived Alumina

Hiroki Fujita, Carlos G. Levi, and Frank W. Zokw

Materials Department, University of California, Santa Barbara, CA 93106-5050

George Jefferson

Materials and Manufacturing Directorate, AFRL, Wright-Patterson AFB, OH 45433

This study focuses broadly on synthesis and characterization ofporous mullite/alumina mixtures for use as matrices in oxidefiber composites. Specifically, an assessment is made of the util-ity of a precursor-derived alumina (PDA) in controlling both themodulus and the toughness of mullite-rich particle mixtures.Property changes are probed through models of the mechanicalbehavior of bonded particle aggregates. Consideration of theconditions needed to cause crack deflection at a fiber–matrixinterface yields an upper allowable limit on the concentration ofPDA to ensure damage tolerance in a fiber composite. The pre-dicted critical concentration lies in the range of about 7–9%,depending on the mullite/alumina ratio in the particle slurry andthe subsequent aging treatment. Values slightly below this limitshould yield composites that exhibit a desirable balance betweenfiber- and matrix-dominated properties.

I. Introduction

DAMAGE tolerance can be enabled in oxide ceramic compos-ites through the use of a controlled amount of fine-scale

matrix porosity, obviating the need for a fiber coating.1–4 Forsuccessful implementation, the matrix must satisfy three criteria:(i) pore retention at the targeted use temperature, (ii) phasecompatibility with the fibers (because of the intimate fiber–ma-trix contact in the absence of a coating), and (iii) adequate me-chanical integrity to ensure acceptable off-axis compositeproperties. When the reinforcements are either alumina or mul-lite/alumina fibers, the requisite matrix characteristics can beachieved through the use of mullite/alumina mixtures. Typically,the mullite is in the form of particles, introduced by infiltrationof a slurry into the fiber preform. Its main role is to form acontiguous nondensifying network. In contrast, the aluminaserves as a binder to impart mechanical strength. It can be in-troduced in one of two ways: by mixing alumina particles intothe slurry, or by subsequent impregnation and pyrolysis of aliquid alumina precursor.5 These lead to two distinct matrixtopologies, illustrated in Fig. 1. The primary objective of thepresent paper is to compare the efficacy of the two approaches incontrolling matrix properties. This is achieved largely through aseries of experiments on porous mullite/alumina compacts withvarying amounts of precursor-derived alumina (PDA) and com-parisons with previously reported measurements on mullite/alu-mina particle mixtures (without precursor additions).6 A secondobjective is to develop and assess models of bonded particle ag-

gregates for use in describing the mechanical properties of com-pacts containing a precursor-derived phase.

The two routes for introducing alumina have three attributes:(i) The use of a mixed mullite/alumina particle slurry allowsboth matrix phases to be introduced simultaneously. By con-trast, the precursor route requires additional steps, beyond thatof slurry infiltration. From a cost viewpoint, the former methodis preferred. (ii) The presence of alumina particles in the slurrycan compromise the stability of the mullite network, especially ifits proportion exceeds the percolation threshold.6 Conversely, ifthe slurry is comprised of only mullite and the alumina is intro-duced subsequently via the precursor route, the contiguity of themullite network is ensured. (iii) The slurry processing route re-sults in an essentially fixed (high) porosity. Consequently, amodest upper limit on the strength of the network is expected. Incontrast, the precursor route allows for filling of the void spacebetween the particles in the network, with attendant elevationsin both the relative density and the mechanical properties. Afurther benefit of the precursor route might be obtained, if,because of capillarity, the PDA forms preferentially at thejunctions of the mullite particles. The present experimentswere designed to address some of these issues.

To motivate the mechanical measurements, a crack deflectionparameter, S, is used, defined by6,7

S � 0:134Gf

Gm

� �1þ Ef

Em

� �0:9

(1)

where G is the toughness, E is Young’s modulus, and subscriptsm and f refer to matrix and fiber, respectively. A matrix crack ispredicted to arrest or deflect at the fiber–matrix interface when Sexceeds a critical value, defined by o � Gi=Gm, where Gi is theinterface toughness. Since the nature of bonding at the fiber–matrix interface is similar to that between particles in the matrix,the interface toughness is expected to be comparable with that ofthe matrix, i.e. o � 1. Thus, upon setting S5o5 1, the criticalcombination of Gm/Gf and Em/Ef is obtained. In light of thisprediction, the subsequent experimental study focuses on thematrix properties Gm and Em. An assessment is then made ofwhether the crack deflection condition is satisfied when a can-didate matrix is combined with the targeted fiber, namely Nextel720.

Despite an extensive literature on the mechanical propertiesof porous ceramics, only a small fraction of the effort has beendevoted to the development and assessment of mechanistic mod-els based on realistic representations of pore topology. Perhapsthe greatest deficiency lies in the low porosity domain, whereinthe materials consist of particle aggregates bonded to one an-other at discrete contact points. In this case, macroscopic prop-erties are extremely sensitive to changes in contact size, evenwhen unaccompanied by changes in macroscopic porosity.6,8,9

Elastic properties of such aggregates have been successfullymodeled by treating the particle junctions as Hertzian contacts

367

JournalJ. Am. Ceram. Soc., 88 [2] 367–375 (2005)

DOI: 10.1111/j.1551-2916.2005.00061.x

K. T. Faber—contributing editor

Funding for this work was provided by the Air Force Office of Scientific Research underContract number F49620-02-1-0128, monitored by Dr. B. L. Lee, as well as a gift fromNGK Insulators.

wAuthor to whom correspondence should be addressed. e-mail: [email protected]

Manuscript No. 10709. Received December 9, 2003; approved August 2, 2004.

and analyzing the aggregate response accordingly.10–13 A similarapproach has been used in conjunction with the discrete elementmethod (DEM) for simulating the fracture process.6 Otherwise,most previous studies have relied on purely empirical relationsto rationalize experimental measurements, usually couched sole-ly in terms of the porosity level.8,14–16 Others have implementedmodels based on rather simplistic (arguably unrealistic) micro-structural representations, e.g. a collection of aligned cylindersof varying diameters, each represented mechanically as a one-dimensional spring with stiffness dictated by cylinder diame-ter.17,18 Although models of this type can be readily analyzed,they lack the realism needed for establishing connections be-tween microstructure and properties. Analogous phenomeno-logical approaches19 have proved to be of similarly limitedutility in modeling fracture energy.

The present article is organized as follows. Section II de-scribes the nature of the materials systems, the processing route,and the measurement procedures. The resulting physical prop-erties and microstructures are presented in Section III. The keymechanical properties—modulus and toughness—are addressedin Sections IV and V. Section VI considers the implications oncrack deflection in fiber composites. For this purpose, the cal-ibrated models are used to establish the critical concentration ofthe precursor-derived phase at which the crack deflection con-dition is no longer satisfied. Conclusions are summarized inSection VII. A comprehensive list of key symbols and their def-initions is presented in Table I.

II. Materials and Test Procedures

Porous mullite compacts, 10 mm� 50 mm� 90 mm in size, werefabricated by vacuum filtration, following procedures detailedelsewhere.5 The particles were MU-107 mullite (Showa DenkoK.K., Tokyo, Japan), with an average diameter of 1 mm, dis-persed in aqueous HNO3 at a pH of 3. Following filtration, thecompacts were dried, sintered at 9001C for 2 h to promote par-tial sintering at the particle junctions, and machined into me-chanical test specimens. Some specimens were subsequentlyimpregnated with an alumina precursor solution, following theprocedures described below, and pyrolyzed at 9001C for 2 h. Allwere subjected to a final aging treatment of either 2 h at 12001C(representing the as-processed condition) or 1000 h at 12001C(to assess thermal stability).

The precursor solution was made by slowly digesting Al pow-der in a solution of AlCl3 � 6H2O in deionized water at 851C. Thefractional volumetric yield of alumina was determined by firingknown quantities of the precursor solution at either 9001 or12001C and subsequently measuring both the mass and the den-sity of the pyrolyzed product. The corresponding densities, 3.48and 3.99 g/cm,3 are consistent with g- and a-alumina, respec-tively. The subsequent yield measurements are reported on thebasis of the g-alumina; the two yields are related by ya/yg5rg/ra � 0.87 where y is the volumetric yield, r is the mass density,and the subscripts refer to the two phases.

The precursor concentration was selected in the followingway. To achieve large porosity reductions with minimal impreg-nation cycles, concentrated solutions are preferred. However, apractical upper limit on the allowable solution concentration isobtained because of solution viscosity and compact permeabil-ity. That is, solution impregnation into the compact at atmos-pheric pressure becomes impractical when the concentration,and hence viscosity, are excessively high. An experimental de-termination of this limit was made by impregnating samples ofknown porosity with solutions of varying concentrations, firingat 9001C, and measuring the weight gain because of the additionof the PDA. The results are plotted in Fig. 2. Initially, the weightincreases linearly with solution concentration. The rate is con-sistent with a prediction based on the measured precursor yield,assuming that all of the pore space within the compact is filledby the solution. Beyond a critical concentration ( � 340 mgAl2O3/cm

3 of solution) corresponding to a precursor yield ofabout 9.5%, the weight gain diminishes rapidly, implying min-imal impregnation. Based in part on these results, two solutionconcentrations were selected for subsequent experiments:yg � 3% and 8%. The latter value lies somewhat below themaximum dictated by permeability. The former value was se-lected to be consistent with that used in previous studies on po-rous matrix composites.3,4 The number of impregnation cycleswas varied from 2 to 10. As shown later, the properties werefound to depend only on the total amount of PDA, independentof the specific combination of precursor concentration andnumber of impregnation cycles.

Previous work on composite fabrication had revealed somebenefits in using a hybrid approach involving both alumina par-ticle additions to the slurry and subsequent precursor impreg-nation and pyrolysis.1,2 The alumina particles were found toimprove the mechanical integrity of the green panels followingthe initial low-temperature heat treatment, relative to that con-taining pure mullite, and allowed handling during subsequentprecursor impregnation without imparting damage to the panel.Furthermore, density measurements on various mullite/aluminaslurries showed some evidence of densification for alumina con-tents exceeding 20%. With this experience, the composites had

Fig. 1. Schematics showing the two matrix topologies produced by (a)mullite/alumina particle mixtures and (b) mullite particles bonded byprecursor-derived alumina.

Table I. Symbols and Definitions

a Junction radiusa0 Initial junction radius (before precursor impregnation)A Surface area per particleA0 Initial surface area per particle (before precursor

impregnation)Ef Fiber Young’s modulusEm Matrix Young’s modulusEs Young’s modulus of solid phaseh0 Height of truncated spherical cap (Fig. A1)n Sintering exponentp0 Initial compact porosity (before precursor

impregnation)p Final porosityt TimetR Reference timeR Particle radiusReff Effective particle radiusVPDA Volumetric concentration of precursor-derived

aluminaV0 Particle volumeya Fractional volumetric yield (a-phase)yc Fractional volumetric yield (c-phase)a Normalize junction size (a/Reff)a0 Normalize initial junction size (a0/R)b Nondimensional precursor ‘‘segregation’’ parameterCf Fiber toughnessCm Matrix toughnessCi Interface toughnessCj Junction toughnessq Mass densityR Crack deflection parameter (Eq. (1))x Toughness ratio (Ci/Cj)

368 Journal of the American Ceramic Society—Fujita et al. Vol. 88, No. 2

been fabricated using a matrix slurry of 80% mullite and 20%alumina (80M/20A), followed by precursor impregnation andpyrolysis. Consequently, the effects of the PDA on the proper-ties of compacts with the latter composition (80M/20A) werealso assessed in the present study.

The open and closed porosities were determined from acombination of Archimedes measurements on the compacts20

and density measurements on the constituent particles using apycnometer. In some cases, the surface area was measured usingthe BET method.21 The microstructures were probed in twoways: by scanning electron microscopy of fracture surfaces, andby energy-dispersive spectroscopy (EDS) of foils in a transmis-sion electron microscope (TEM).

Young’s modulus of each fired compact was measured usingstrain-gauged flexure specimens, 5 mm� 5 mm� 50 mm long,loaded in four-point bending. The inner and outer loading spanswere 20 and 40 mm, respectively. Toughness was obtained fromthree-point bend tests on chevron-notched specimens with thesame dimensions and instrumented with an LVDT to measureload-point displacement. In cases where the crack grew stablyacross the specimen cross-section, the toughness G was calcu-lated from the work of fracture.22 In the few instances wherefracture occurred catastrophically at the load maximum, thefracture toughness, Kc, was calculated from the peak load andthe stress intensity solutions developed by He and Evans.23 Itwas then converted to G through the Irwin relation using theaverage measured Young’s modulus from the unnotched tests.Figure 3 shows comparisons of the results from the two methodsfor cases in which stable growth had occurred. In the vast ma-jority of cases, the two values are within about 30% of one an-other. This difference coincides closely with the variability in themodulus measurements for a given material, indicating that thetwo measurement techniques are consistent with one another.

III. Physical Properties and Microstructures

The effects of precursor impregnation and pyrolysis on compactporosity are summarized in Fig. 4. Also shown for comparisonis a prediction, assuming that: (i) all of the pore space is filled bythe precursor solution on each impregnation cycle; (ii) duringpyrolysis (9001C), the precursor solution is converted to g-alu-mina; and (iii) during the final heat treatment (12001C), theg-alumina is converted to a-alumina. The predicted porosity isgiven by

p

p0¼ 1� ð1� 1� yg

� �NÞ rgra

(2)

where p0 is the porosity of the compact before impregnation andN is the number of impregnation cycles. Upon fitting Eq. (2) tothe experimental data, the volumetric yields of the two precur-sors are inferred to be 2.4% and 6.7%. These are about 20%lower than the values obtained from the precursor solutionsalone (3% and 8%). The discrepancy may arise from one ormore sources. Losses of precursor solution from the compactmay occur: notably, during the period between the completionof precursor impregnation and the onset of gelation. While nosignificant closed porosity was detected in the final compacts, itis possible that closed porosity could have arisen during theconversion of the dried precursor to transient alumina. In thatevent, the net amount of open porosity susceptible to impreg-nation would decrease, and hence the amount of aluminathat can be incorporated in each impregnation cycle. This ‘‘tran-sient’’ closed porosity could be subsequently reopened byshrinkage of the precursor-derived oxide during the final con-version to corundum, and/or densified in part as the typicallynanocrystalline transient alumina is heated to higher tempera-tures during the conversion stage.

Representative EDS maps of two prototypical microstruc-tures are shown in Fig. 5. One is of the 80%M/20%A particlemixture (without precursor addition), and the other comprisesparticles of pure mullite and about 15% PDA, the latter comingfrom 10 impregnation cycles of the concentrated precursor. Themaps are produced by separately mapping out regions contain-ing Al and Si and then superimposing the two images in such a

Fig. 2. Effects of precursor solution concentration on viscosity and, inturn, on the degree of solution impregnation into a porous matrix com-posite. Solid line shows the predicted trend in weight gain assuming all ofthe pores within the matrix are filled by the solution.

0.1

1

10

100

0.1 1 10 100

Γ (J

/m2 )

from

pea

k lo

ad

Γ (J/m2) from work of fracture

Fig. 3. Comparisons of toughness values determined from work offracture and from the peak load in chevron-notched bend specimens.

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 8 10 12

Rel

ativ

e P

oros

ity ,

p/p o

Number of Impregnations, N

Concentrated precursor:yγ = 6.7%

Dilute precursor:yγ = 2.4%

642

Fig. 4. Reduction in porosity following precursor impregnation andpyrolysis. Solid lines are fits of Eq. (2) to the experimental data.

February 2005 Porous Mullite/Alumina Mixtures 369

way that alumina appears red and mullite appears blue. In theformer case, the distinct nature of the two particle types andtheir distribution are evident; in the latter, the alumina forms acoating (albeit somewhat nonuniform) on the mullite particles,roughly consistent with the schematic shown in Fig. 1(b).

The results from a series of surface area measurements show-ing the effect of the PDA are plotted in Fig. 6. Here the specificsurface area is normalized by the value, S0, prior to precursorimpregnation. The results show a reduction in specific surfacearea with precursor addition, but generally lower than that ex-pected from simple geometric models based on randomlypacked spherical powders. Two extreme scenarios are analyzed(Appendix A). One assumes that the precursor-derived oxide isuniformly distributed over the surfaces of the powders, buildingup the necks as its thickness increases (Fig. A-1(a)). The otherconsiders the precursor to segregate completely to the inter-par-ticle contacts, with no material added to the surface of the pow-ders away from the necks (Fig. A-1(b)). The models allow forthe presence of an initial neck size, produced during the matrix-strengthening treatment prior to the first precursor impregna-tion. These two scenarios, albeit idealized, provide usefulbounds to the reduction of pore surface area with increasingdensification, as well as to the rate of buildup of the necks be-tween particles (Fig. 6(b)). Comparison with the experimentalresults in Fig. 6(a) reveals that, in the early stages of the process,the normalized area falls within these bounds and is closer to theuniform distribution model. There is, however, evidence thatsome amount of segregation occurs, as discussed in Section IV.The implication is that the experimental points should continue

to fall within the boundaries as densification proceeds, but inreality the pore area decreases at a slower rate than that expect-ed from either model.

Further insight into the sources of the above discrepancy canbe obtained from the fracture surfaces of the broken porouscompacts (Fig. 7). For moderate PDA additions, the fracturesurfaces are seemingly indistinguishable from those of the par-ticle aggregates alone; compare, for example, Fig. 7(a) and (b)with (c) and (d). This result suggests that the coating is thin anduniform, in qualitative agreement with the specific surface areameasurements for low PDA loadings in Fig. 6. For larger PDAadditions (Fig. 7(e) and (f)), the PDA coating breaks up intosmall discrete islands on the larger mullite particles. Such islandformation leads to an increase in specific surface area and hencea lower rate of pore surface reduction with densification thanthat expected from the model predictions in Fig. 6(a).

IV. Young’s Modulus

(1) Experimental Measurements

The modulus measurements are plotted in Fig. 8 as a functionof the percentage of total volume occupied by the PDA:VPDA5 p0�p. In this context, the PDA is extremely effective.For instance, a 10% addition of PDA to a pure mullite particleaggregate leads to about a 10-fold increase in modulus. In con-trast, the particulate alumina in the slurry plays only a minorrole; the results for the two slurry compositions—100%M and80%M/20%A—are similar to one another.

Fig. 5. Energy-dispersive spectroscopy/transmission electron micro-scope maps of thin foils of (a) 80% mullite/20% alumina (without pre-cursor addition) and (b) pure mullite compact impregnated 10 times withthe concentrated precursor, yielding VPDA5 15.8%.

Fig. 6. Effects of precursor-derived alumina on (a) normalized specificsurface area and (b) neck size. Details of the model predictions are givenin the Appendix A. (Uncertainty in each measurement in (a) is less thanthe symbol size.)


The effects of the aging treatment are most pronounced in theparticle compacts that are free of the PDA. For instance, forVPDA5 0, the modulus of the pure mullite compact increases bya factor of about 3 during aging, despite the absence of meas-urable density change.6 The change has been attributed to sin-tering at the particle junctions by a (nondensifying) surfacediffusion mechanism.6 The large relative increase in modulusresults from the very small junction size (and hence modulus) inthe starting material such that even modest amounts of junctiongrowth yield large increases in modulus. Similar trends are ob-tained in compacts of 80%M/20%A. The relative magnitude ofthe sintering effect becomes progressively smaller as VPDA in-

creases. Indeed, for VPDA � 8–10%, the effect is overshadowedby statistical variations in the measurements. This implies thatthe increase in modulus because of the PDA is much greaterthan that associated with sintering.

(2) Simulations and Modeling

Effects of the PDA on the modulus of the compacts formedfrom 100% mullite particles have been computed using simula-tions based on the DEM.13 In this method, the material ismodeled as a collection of particle elements. The junction re-sponse that defines the particle element properties is derivedfrom a finite-element analysis (FEA) of a single particle in aperiodic array. The interaction between particle junctions, char-acterized by the displacement of one junction because of theforce acting on another, is also derived from the FEA. Next, anisotropic random aggregate of touching spherical particles isgenerated using a computer algorithm. To ensure equilibrium,each particle is required to touch at least three neighbors uponplacement onto the existing aggregate. The final particle packingdensity is about 55% and the average coordination number isabout 6. For monophase systems, junction growth is simulatedby uniformly expanding the particles and redistributing theoverlapping material uniformly over the free surface of the par-ticles. For the two-phase system comprising a contiguousparticle network and a precursor-derived phase, the material ismodeled as an aggregate of touching particles, all coated with auniform layer, thickness h, of the second phase. This idealizationis depicted in Fig. A1. The analyses described in the Appendix Ashow that the coating thickness can be related to the normalizedjunction size a and the volume fraction of PDA. The predictedvalues are plotted in Fig. 6(b) for a coordination number of 6and two values of the initial neck size (a05 0, 0.1). To first or-der, the junction size is given by

a �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2VPDA

3 1� p0ð Þ þ a20

s(3)

For the two-phase system, the elastic response of the junc-tions is calculated using a periodic array FEM model, with thetwo phases explicitly discretized. For both mono- and two-phasesystems, the random particle network is then subjected to a pre-scribed macroscopically uniform strain field and the effectiveelastic response is determined using the DEM.13 In the presentcalculations, Young’s moduli E and Poisson’s ratios n of thetwo phases are taken to be EA5 400 GPa, EM5 200 GPa, andnA5 nM5 0.2, where the subscripts A and M refer to aluminaand mullite, respectively.

Figure 9 shows the results of these simulations for the two-phase system (with a05 0) as well as for monophase aggregates

Fig. 7. Fracture surfaces of three representative porous compacts, allwith pure mullite particle slurry, but with varying amounts of precursor-derived alumina (PDA): (a, b) VPDA5 0; (c, d) VPDA5 8.3%; (e, f)VPDA515.8%. All had been aged for 2 h at 12001C.

Aging time

2h 1000h

100M80M/20A

0

50

100

150

0 5 10 15 20

You

ng's

mod

ulus

, E (

GP

a)

PDA concentration, VPDA (%)

αo = 0, β = 1αo = 0.1, β = 1.2

1000 h

2 h

Model: Mullite

Fig. 8. Summary of Young’s modulus measurements and model pre-dictions. The predictions are based on the results of the discrete elementsimulations, shown in Fig. 9 and described in the text. Strictly, thesepredictions pertain only to the pure mullite slurries. Consequently, validcomparisons can only be made between the two solid lines and the twosets of filled symbols.

0

10

20

30

40

50

0 0.1 0.2 0.3

You

ng's

mod

ulus

, E (

GP

a)

Normalized junction radius, α = a / (R+h)

Mullite particles

DEM Eqn. 4Eqn. 7

Alumina particles

Mullite particleswith alumina coating

Fig. 9. Discrete element method (DEM) simulations of Young’s mod-ulus of monophase particle aggregates (either alumina or mullite) andmullite particle aggregates with a uniform coating of alumina.


of pure mullite and pure alumina, plotted against a. The latterresults are well described by the empirical formula

�E

Es¼ 0:4a 1þ 14 a3

� �(4)

where Es is the modulus of the solid. Moreover, the modulus ofthe two-phase aggregate can be estimated by the volume frac-tion-weighted average of the moduli of the two monophase ag-gregates, evaluated at the same value of a. This estimate isobtained by replacing Es in Eq. (4) with the volume-weightedaverage of the moduli of the two solid phases. Upon calculatingthe volume ratios from geometry, the estimated modulus of thetwo-phase system becomes

�E ¼ 0:4a 1þ 14 a3� �

EMuM þ EA 1� uMð Þ½ � (5)

where

uM ¼ 1� p0

1� p0 þ VPDA(6)

Upon calculating VPDA as a function of a from Eq. (3), witha05 0, and combining Eqs. (4), (5), and (6), the estimated mod-ulus of the two-phase system becomes

�E ¼ 0:4a 1þ 14 a3� �

EM1� p0ð Þ þ EA=EMð ÞVPDA

1� p0ð Þ þ VPDA

� (7)

The modulus estimates for the mullite/alumina system ob-tained by this averaging procedure, shown by the dashed line inFig. 9, agree well with the results of the DEM simulations.

The model results are replotted in Fig. 8, for comparison withthe experimental measurements. Several features are notewor-thy. Because the junctions between the mullite particles aremodeled as point contacts when a05 0, the predicted modulusgoes to zero when VPDA5 0. This behavior differs from thatobtained experimentally. That is, the modulus of the mulliteparticle aggregate is finite, at about 10 GPa: a consequence ofthe finite (but small) junction size. The predicted modulus sub-sequently increases with VPDA, but at a rate that is significantlylower than that of the measurements. The rather large discrep-ancy at large values ofVPDA (approaching 60 GPa) suggests thatother sources of error must be present, apart from the neglect ofthe initial junction size. It is surmised that, because of capillarity,the PDA forms preferentially at the mullite particle contacts,rather than as a uniform particle coating, thereby stiffening theparticle junctions and elevating the composite modulus.

The effects of both the initial junction size and the nonuni-formity of the PDA can be taken into account through an ap-proximate modification to the DEM results. For this purpose,Eq. (3) is modified to read as

a0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

2VPDA

3 1� p0ð Þ þ a20

s(8)

where b is a nondimensional parameter, � 1, that accounts forpreferential alumina formation at the particle junctions. In thiscontext, the normalized junction size a in both Fig. 9 and Eq. (8)is reinterpreted as a0. Combining Eqs. (7) and (8) then yields thevariation in �E with VPDA. The resulting predictions for b5 1.2and a05 0.1 are shown in Fig. 8. This combination yields a goodfit to the data for the 2 h aging over the entire range of VPDA.The corresponding buildup in neck size is compared in Fig. 6(b)with those resulting from the two extreme scenarios analyzed inthe Appendix A. It is noted that the inferred level of segregation,reflected in the value of b, is modest. The neck size curve inFig. 6(b) is closer to the uniform distribution scenario thanto the full segregation model, in qualitative agreement withthe specific area measurements discussed in the context of

Fig. 6(a). Furthermore, although this prediction is strictly ap-plicable to the compacts containing pure mullite particles, it ap-pears to underestimate only slightly the results for the 80M/20Amixtures.

The effects of the extended (1000 h) aging treatment on themodulus of compacts containing the PDA are treated in thefollowing way. In an approximate sense, the addition of PDAcan be viewed as being equivalent to a sintering treatment, pro-vided both yield the same changes in junction size without af-fecting interparticle spacing. This equivalence stems from thefact that the modulus is dictated by the current junction size,essentially independent of the route taken to achieve it (not-withstanding slight differences in the moduli of the two phases).In the sintering case, the junction size evolves according to apower law of the form:6

a ¼ t

tR

� �1=n

(9)

where t is the sintering time, tR is a reference time, and n is aconstant, the latter two depending on the transport mechanism.(For mullite, junction growth occurs by surface diffusion,whereupon n5 7.6) The junction size a0 that results from pre-cursor addition (according to Eq. (8)) can be viewed as havingbeen developed through a sintering treatment of the monophaseparticle aggregate, for an equivalent sintering time, teq, given by

teq ¼ tRða0Þn (10)

After additional sintering for time t, the final junction size isevaluated from Eq. (9), using the total sintering time, t1teq. Theresult is

a ¼ a0 1þ t

tR

� �1

a0

� �n� 1=n(11)

The model prediction (Eqs. (7) and (11)) is plotted in Fig. 8,using a reference time tR5 2� 106 h. This adequately capturesthe trend for the pure mullite particle aggregates with addedPDA. It is of interest to note that the corresponding referencetime inferred from sintering treatments of pure mullite (absentPDA) is tR5 8� 106 h:6 the fourfold difference reflecting thehigher sinterability of the alumina in the PDA-containing mix-tures. Indeed, it is for this reason that the present model pre-dictions (using the lower value of tR) slightly overestimate themodulus for the pure mullite (VPDA5 0).

In light of the rather weak sensitivity of the modulus to theslurry composition (100M vs 80M/20A), no attempt was madeto explicitly model a two-phase particle aggregate containing aprecursor-derived phase. Models for the behavior of the two-phase aggregate alone are presented elsewhere.6

V. Toughness

The toughness measurements are plotted in Fig. 10. For the as-processed materials, the toughness increases approximately lin-early with VPDA, almost independent of the slurry composition(100M vs 80M/20A). Furthermore, the toughness increasesslightly following the additional 1000 h aging. These resultsare consistent with the associated increases in junction size be-cause of both addition of PDA and sintering.

Also shown in Fig. 10 are the predictions of numerical sim-ulations of fracture, again using the DEM. The results of thesimulations for monophase aggregates are well described by:6

GGj

¼ 12a2 (12)

where Gj is the intrinsic junction toughness. This result is adapt-ed to the PDA-containing aggregates by using the junction size


given in Eq. (8), whereupon the toughness becomes

GGj

¼ 12 b22VPDA

3 1� p0ð Þ þ a20

� �(13)

For consistency with the modulus predictions, simulations forthe 2 h aging treatment were performed using a05 0.1 andb5 1.2, with Gj being a fitting parameter. This procedure yieldsGj5 4 J/m2: only slightly greater than the value (3 J/m2) that hadbeen inferred previously from measurements on pure mullite.6

The effects of extended aging are taken into account in thesame manner described above for the modulus. That is, junctiongrowth is assumed to follow in accordance with Eq. (11), with areference time, tR5 2� 106 h. Combining this result withEq. (12) yields the predicted curve in Fig. 10. Although the pre-diction yields the correct qualitative trends, the scatter in theexperimental measurements precludes a more critical assessmentof the model. Indeed, some discrepancy is expected, since thepertinent reference time should depend on the amounts of thetwo phases present at the particle junctions, as dictated bythe PDA concentration.

VI. Crack Deflection

The measured values of modulus and toughness have been usedto compute the crack deflection parameter S (Eq. (1)), assuming

that the aggregates are combined with Nextel 720 fibers to forma composite. The results are plotted against VPDA in Fig. 11.Here, again, the two slurry compositions yield similar results.For the as-processed materials, S decreases with increasingVPDA and eventually falls below the critical value, o � 1, atVPDA � 9%. This point is expected to mark the onset of crackpenetration into the fibers and essentially a complete loss indamage tolerance. Extended aging causes a reduction in S, es-pecially at low values of VPDA. Furthermore, it reduces the crit-ical value associated with crack penetration, to VPDA � 7–8%.

The predicted results from the models for modulus andtoughness (calibrated accordingly) are also shown in Fig. 11.A satisfactory correlation with the experimental measure-ments is obtained. Most notably, the predicted critical value,VPDA � 9–10%, is broadly in line with that inferred from themeasurements.

VII. Conclusions

The mechanical properties of mullite particle aggregates can beeffectively tailored by the addition of PDA. Tenfold changes inboth modulus and toughness can be obtained through the ad-dition of moderate amounts of alumina, in the range of 10–15%.Significantly smaller changes can be effected through the use ofalumina particle additions. The precursor-derived route has theadditional advantage over the mixed mullite/alumina particleroute in that the contiguity of the mullite particle network is notcompromised. Consequently, the structure is relatively immunefrom densification at the targeted upper use temperature for thecorresponding fiber composites.

The mechanical properties of the two-phase aggregates can bedescribed using idealizations of the particle and precursor to-pology coupled with simulations based on the DEM. Compar-isons between predicted and measured properties suggest thatthe precursor-derived phase exhibits some degree of segregationto the particle junctions, yielding larger property elevations rel-ative to that for a uniform precursor distribution.

Limits have been established on the maximum acceptableconcentration of the precursor-derived phase, such that a com-posite with the candidate matrix would satisfy the crack defec-tion condition. In order to produce composites that are damagetolerant under fiber-dominated loadings yet retain adequate off-axis properties, precursor concentrations slightly below the crit-ical value would be preferred. Experiments are in progress toassess these predictions.

Appendix A: Models for Precursor Buildup on Particles

(A.1) Preliminaries

Models representing two extreme scenarios of precursor builduparound the matrix particles are developed. The geometrical fea-tures of the models are depicted in Fig. A1. The goal is to de-scribe the variation in specific surface area (defined as pore areaper unit mass or particles) and contact area between particles asthe relative density increases with the addition of precursor-de-rived ceramic. The particles are assumed to be initially sphericalwith radius R and packed in a random configuration with co-ordination number z. The models include the possibility of a fi-nite contact area of radius a0, developing during the initial heattreatment, prior to precursor impregnation. In that event, theinitial surface area and volume per particle are

A0 ¼ 4pR2 1� z

2

h0

R

� �� (A-1a)

V0 ¼4

3pR3 1� z

4

h0

R

� �2

3� h0

R

� �" #(A-1b)

where h0 is the height of the spherical cap that is truncated fromthe particle to create an initial neck of radius a0 (Fig. A1).

Aging time

2h 1000h

100M80M/20A

0

5

10

15

20

0 5 10 15 20

Tou

ghne

ss, Γ

(J/

m2 )


αo = 0.1, β = 1.2

Model: Mullite

1000 h

2 h

Fig. 10. Summary of toughness measurements and model predictions.The model predictions are based on the parameter values used to cal-culate Young’s modulus in Fig. 8.

0.1

1

10

100

0 5 10 15 20

Cra

ck d

efle

ctio

n pa

ram

eter

, Σ


Σ = 1

Aging time

2h 1000h

100M80M/20A

αo = 0.1, β = 1.2

Model: Mullite

1000 h

2 h

Fig. 11. Effects of precursor-derived alumina (PDA) concentration andaging on the crack deflection parameter, S, calculated using fiber prop-erties Gf5 15 J/m2 and Ef5 260 GPa.


In general, the relative (specific) surface area is given by theexpression

S

S0¼ A

A01þ rA

r0

� �DVV0

� �1

(A-2)

where r0 and rA are the densities of the particles and the PDA,respectively, and A and DV are the pore area and volume ofPDA added per particle, respectively. The porosity is given by

p ¼ p0 � 1� p0ð ÞDVV0

(A-3)

where p0 is the initial porosity. The normalized neck radius isdefined as

a ¼ a

Reff(A-4)

where Reff is the current radius of the particle away from theneck area. Its initial value, before precursor impregnation, is

a0 ¼2h0

R� h0

R

� �2 !1=2

(A-5)

(A.2) Uniform Distribution

In one scenario the PDA is assumed to be distributed uniformlyover the free surface of the particles as a coating with thicknessh, except where prohibited by particle contact (see Fig. A1(a)).The analysis is based on a single representative particle and itscorresponding coating and void space. The pore surface areaand PDA volume per particle are given by

A ¼ 4pR2

(1� z

2

h0

R

� �þ 2� z

21þ h0

R

� ��

� h

R

� �þ 1� z

2

� h

R

� �2) (A-6a)

DV ¼ 4

3pR3 1þ h

R

� �3

�1� z

42

h

R

� �3

þ3 1þ h0

R

� �"(

� h

R

� �2

þ6h0

R

� �h

R

� �#)

(A-6b)

The porosity is related to the normalized thickness h/R throughEqs. (A-3) and (A-6b). For this geometry Reff5R1h and thenormalized neck radius is

a ¼ 2 1þ h

R

� �h0

Rþ h

R

� �� h0

Rþ h

R

� �2" #1=2

� 1þ h

R

� ��1

(A-7)

Combining this with Eq. (A-5) yields

a ¼ a20 þ2h

Rþ h

R

� �2" #1=2

1þ h

R

� ��1

(A-8)

(A.3) Precursor Segregation to Necks

In a diverging scenario, the precursor is assumed to be fullysegregated to the necks. For simplicity, the PDA buildup is tak-en to have a cylindrical geometry as illustrated in Fig. A1(b).The half thickness at the edge of the neck is designated as k andis analogous but not equivalent to the h used in the earlier case.The pore surface area and PDA volume per particle are nowgiven by

A ¼ 4pR2

(1� z

2

h0

R

� �� z

2

k

R

� �

� 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih0

Rþ k

R

� �2� h0

R� k

R

� �s" #) (A-9a)

DV ¼ 4

3pR3 z

4

k

R

� �2

3 1� h0

R

� �� 2

k

R

� �� ( )(A-9b)

As before, the porosity is related to the normalized thicknessk/R through Eqs. (A-3) and (A-9b). For this geometry, Reff5R and the normalized neck radius is

a ¼ 2h0

Rþ k

R

� �� h0

Rþ k

R

� �2" #1=2

(A-10)

Combining with Eq. (A-5) yields

a ¼ a20 þ2k

R

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a20

q� k

R

� �2" #1=2

(A-11)

Comparison of the models is performed by setting a value ofp0 and then calculating the values of h/R and k/R implicitly fromthe combination of Eq. (A-3) and either Eq. (A-6b) or (A-9b).Once the pertinent value of the normalized thickness is known,S/S0 and a can be readily calculated. The results of such calcu-lations are compared with the experimental measurements inFig. 6.

Fig.A1. Schematics of the two extreme scenarios in precursor topolo-gy: (a) uniform distribution on the particle surfaces, and (b) completesegregation to the particle necks.


Acknowledgments

The authors gratefully acknowledge the assistance of J. Y. Yang (UCSB) inmaterials processing and theMaterials Research Laboratory of NGK Insulators inthe TEM/EDS analysis.

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Controlling Mechanical Properties of Porous Mullite/Alumina