PHYSICAL REVIEW 8 VOLUME 48, NUMBER 21 1 DECEMBER 1993-I

Raman studies of sol-gel alumina: Finite-size effects in nanocrystalline Alo(OH)

C. J. Doss and R. ZallenDepartment of Physics, Virginia Tech, Blacksburg, Virginia 24061

(Received 17 August 1992; revised manuscript received 30 August 1993)

A systematic Raman-scattering investigation has been carried out on sol-gel alumina prepared by thehot-water hydrolysis and condensation of Al(OC4H9)3, the Yoldas process, as a function of process vari-ables such as the time spent in the sol phase. Nanocrystalline boehmite, y-A1O(OH), is the principalcomponent of these materials. We have found small but systematic changes, as a function of sol agingtime, in the line shape and position of the dominant boehmite Raman band observed in the alumina hy-drogels. These spectral changes are interpreted in terms of nanocrystallinity-induced finite-size effectsassociated with the slow growth of A10(OH) nanocrystals in the sol. X-ray-diffraction experiments wereused to determine nanocrystal sizes (as small as 3 nm for gels prepared from fresh sols) and to estimategrowth kinetics from the Raman-line-shape results. The Raman peak-position shift is proportional toI. , where I. is the average nanocrystal size and a is a Raman-versus-size scaling exponent. ForAlO(OH) we And n to be 1.0, close to the scaling-exponent values reported for graphite and boron ni-tride (BN) and different from the values (about 1.5) that describe the reported behavior of Si and GaAs.

I. INTRODUCTION

A sol-gel route to porous transparent alumina was es-tablished by Yoldas in 1975 in a series of widely cited pa-pers. ' Sol-gel alumina, prepared by the Yoldas pro-cess, is the prototypical example of a crystalline (meaningmicrocrystalline or nanocrystalline) gel. Figure 1

presents a schematic of the synthesis procedure, as imple-mented in the present study. Since our study addressesthe sol-gel aspects and the structure of the resulting gels,we omit the subsequent thermal treatments used to con-vert the gels to porous, yet transparent, alumina monol-iths. '

The Yoldas process for sol-gel alumina includes threemain steps: (i) hydrolysis and condensation of an alumi-num alkoxide in excess water to form an A10(OH) pre-cipitate; (ii) resuspension and dissolution of the precipi-tate by acid peptization to form a clear, colloidal sol; (iii)concentration of the sol by solvent boiloft; resulting in theformation of a gel.

The initial hydrolysis step is done with hot water, andthe peptized sol is kept hot in order to form A10(OH)sols and gels rather than precipitates of Al(OH) 3

phases. ' Alumina gels are mainly water and shouldproperly be referred to as alumina hydro gels.Throughout this paper, whenever the term gel appearswithout a modifier, alumina hydrogel is meant. Driedgels will be specifically referred to as xerogels.

In this paper, we report the results of a systematicRaman-scattering investigation of sol-gel alumina materi-als as a function of process parameters, particularly thesol aging time (t, in Fig. 1). Nanocrystalline boehmitey-A10(QH), is shown (by Raman and x-ray measure-ments) to be the principal nonaqueous component and wehave found small but systematic changes, as a function oft„ in the line shape and position of the dominant boeh-mite Raman band. These spectral changes have been in-

terpreted in terms of nanocrystallinity-induced finite-sizeeffects associated with the slow growth of A10(OH) nano-crystals in the sol phase, and we have used them to deter-mine estimates of characteristic nanocrystal sizes andgrowth kinetics in this system. This interpretation is sup-ported by x-ray-diffraction experiments (which providethe nanocrystal-size calibration for the Raman changes)and is consistent with crystal-growth models.

Section II describes the experimental techniques usedin the synthesis of, and measurements on, the sol-gelalumina materials. The Raman spectrum of crystallineboehmite is described in Sec. III, as a necessary preludeto the measurements on the gels. Raman scattering fromalumina gels, and the observation of systematic changesaccompanying sol-gel processing, are described in Sec.IV. The interpretation in terms of nanocrystallinity and

{two liquids)

1 mole

AI(OC4H9) 3

I

hydrolysis 8condensation

110 moles

H~O (80 C)

I

{fast)

AlO{OH)

precipitate

20min

AIO(OH) gel

(ti )

boil toconcentrate

AIO(OH) gel

(t )

Ji

AIO(OH) gel

(t. )

&k

V

peptize95 C

with acid

{colloidal sol)

40 min AIO(OH) sol, 1 hour AIO(OH) sol AIO(OH) sol

t, =1 hour t,=2 hours t,=s hours

FICx. 1. Schematic of our implementation of the Yoldas pro-cess for synthesizing sol-gel alumina. After time t, in the solphase at 95 C, gelation was brought on by boiloff-induced con-centration. Raman studies were done on hydrogels prepared us-

ing various sol aging times (t, ), and the effect of varying theacid peptization step in the synthesis was also studied.

0163-1829/93/48(21)/15626(12)/$06. 00 15 626 Qc1993 The American Physical Society

48 RAMAN STUDIES OF SOL-GEL ALUMINA 15 627

finite-size effects is given in Sec. V. X-ray-diffractiondetermination of nanocrystal sizes is described in Sec.VI, and results for the kinetics of nanocrystal growth aregiven in Sec. VII. Section VIII is a summary of our prin-cipal findings.

II. EXPERIMENT

A. Preparation of alumina gels

The Yoldas process for making clear alumina gels, asindicated in Fig. 1, consists of the three main steps listedin the Introduction. The alkoxide we used, aluminumsecondary butoxide (ASB), Al(OC4H9)3, was doubly dis-tilled and stored in an inert atmosphere chamber. ASB isa clear, viscous liquid at room temperature. It was heat-ed to reduce its viscosity, drawn into a hypodermicsyringe, and added by syringe to the open reaction vesselcontaining water at 80 C under vigorous stirring condi-tions. The H20:ASB molar ratio was 110. ASB additioninduces immediate hydrolysis and condensation reactionsin the water-rich mixture ' ' and yields an A10(OH) pre-cipitate.

The initial hydrolysis temperature of 80'C was chosenbecause it is the highest temperature compatible withavoiding violent boiling caused by the exothermic hydro-lysis and condensation reactions. Following the ASB ad-dition, the temperature controller was set to 95 C. Ittook about 20 min for the liquid to reach 95'C, partly be-cause of the boiloff of the alcohol (secondary butanol)formed as a by-product of the hydrolysis reaction. Twen-ty minutes after hydrolysis and condensation, acid wasadded to cause peptization and the dispersion of the pre-cipitate to form a clear, stable, colloidal sol. Twodifferent acids (0.1 N HNO3 and 0.1 N HC1) and twodifferent acid concentrations (R =0.07 and R =0. 14,where R is the acid:ASB mole ratio) were used in varia-tions of the peptization procedure.

Following acid addition, the reaction vessel was closed(except for a very small syringe needle hole to equalizepressure) to prevent solvent evaporation. The sol wasthen maintained, with continual stirring, at 95'C. Atselected times, a sol sample was removed by pipette fromthe reaction vessel and boiled (for about 10 min) in asmall open beaker to remove water and concentrate.With sufficient water removal, the sol becomes viscousand approaches the gel point. The viscous sol was thentransferred to a glass vial, sealed to retain the remainingwater, and allowed to cool to room temperature. It thengelled within a few minutes.

These hydrogels are about 90% water, by weight.Early gels (gels made from sols aged for only a few hours)are slightly cloudy; later gels (made from sols aged formany hours) are clear. The cloudiness is attributed to in-complete peptization for short sol aging times. Once agel is formed, its degree of cloudiness or clarity does notchange, nor does its Raman spectrum. As described inSec. IV, Raman studies reveal structural differences inthe gels, which depend on the amount of time spent bythe parent sol in the reaction vessel.

B. Preparation of related materials

A series of samples were prepared with a proceduresimilar to that of Fig. 1, with the key difference of omit-ting the acid addition peptization step. These materialsremain as precipitates, in contact with the hot, stirred,aqueous bath in the form of a cloudy suspension. Atselected times, a sample of the suspension was removedand centrifuged to remove most (about 75%%uo) of the wa-ter. The resulting white mush was then oven dried at 110'C for 24 h, yielding brittle, white chunks. Raman stud-ies of these dried precipitates (Sec. IV) reveal that theirstructure, like that of the gels, also depends on history,the time spent in the reaction bath.

In addition to the hydrogels and the precipitates, a fewother samples were investigated. Portions of some of thehydrogels were subjected to a 110 C oven drying step(like that used for the precipitates), removing their watercontent and yielding xerogels in the form of transparentchunks. An unpeptized precipitate (referred to as a130'C precipitate) powder sample was made by addingASB to H20 (ASB:HzO molar ratio was 110) and process-ing the sol in a pressure vessel at 130'C for about fivedays. The sol was then dried in an oven at 110 C. Theproduct of this process was a white powder. A sample ofcrystalline boehmite powder (composed of micron-sizecrystals), prepared by autoclaving aluminum metal in ex-cess water, was provided by Alcoa laboratories. X-raydiffraction confirmed that the material was well-crystallized boehmite.

C. Raman-scattering experiments

Raman spectra (Stokes spectra) were obtained at roomtemperature, using a SPEX 1403 Raman spectrometerwith a GaAs photocathode photomultiplier and photon-,counting electronics. A cw argon-ion laser, operating at488.0 or 514.5 nm, was the excitation source. The laserpower at the sample was approximately 80 mW. We usedan instrumental bandwidth of 3.8 cm ' and a 0.5 cmgrating step size. Spectra were obtained in the 90'scattering configuration, collecting both polarizations ofscattered light. For the hydrogels, spectra were takenwith the gel held in its vial, laser light entering from thebottom and scattered light collected through the side(each sample was scanned for a total of 40 min). Thebeam focus was inside the gel; no spectral features attri-butable to the glass vial were observed. Xerogels andprecipitates were measured in air, glued onto a capillarytube. The crystalline boehmite powder and the 130'Cprecipitate powder were held in capillary tubes for Ra-man measurements.

The experimental Raman linewidth is a convolution ofthe actual linewidth I and the instrumental bandwidthr,„.t, via r,„pt=(r2+r2..t)1/2 In the analysis presented, s

we use the (deconvoluted) results for the estimated actuallinewidth I .

D. X-ray-diAraction experiments

X-ray-diffraction traces were obtained at room temper-ature, using a Scintag XDS-2000 powder diffractometer

15 628 C. J. DOSS AND R. ZALLEN 48

and Cu Ka radiation with a wavelength of 1.54 A. Thex-ray measurements were carried out on powder samples(prepared by gentle grinding of the dried precipitate orhydrogel) placed on an oriented quartz substrate.DifFraction done directly on hydrogel films cast on thesubstrate missed several boehmite peaks because ofdrying-induced preferred orientation, so powder sampleswere prepared by room-temperature drying of the hydro-gels, followed by gentle grinding.

Crystallite sizes were determined by the Scherrer equa-tion, "

( g)0.94KI. cosO

'

where L is the microcrystallite size, 0 is the peak posi-tion, and B(28) is the linewidth. Linewidth correctionsfor x-ray instrumental broadening were performed in thesame manner as they were for the Raman linewidths. '

III. RAMAN SPECTRUM OF BOEHMITE [AIO(OH)]

which has intralayer-shear bond-bending character, but adefinitive assignment awaits a more detailed force-fieldanalysis.

Figure 3 presents Raman-scattering results obtainedfrom boehmite powder composed of micron-scale crystal-lites. ' At these relatively large particle sizes, the ob-served spectrum should be indistinguishable from that ofthe bulk crystal. This is supported by the sharpness ofthe Raman lines in Fig. 3. (An x-ray-diffraction spectrumfor this sample, displaying sharp lines, is shown in Sec.VI.)

The spectral region shown in Fig. 3 contains thelattice-mode regime, i.e., the modes are predominantlyAl-0 vibrations. ' Table I lists the peak positions,linewidths, and relative intensities of the Raman lines ofboehmite. The linewidth given is the full width at halfmaximum (FWHM), after correcting for the instrumentalbroadening. The sharp, intense line at 363 cm is thedominant spectral signature of boehmite. The line-shape

Boehmite, y-AIO(OH), is the predominant phasepresent in the alumina gels formed by the Yoldas processof Fig. 1. It is, therefore, necessary to consider first theRaman spectrum of this crystal form. The crystal struc-ture of boehmite is shown in Fig. 2."' The structure isorthorhombic, space group Cmcm (DzI, ), with twoA10(OH) formula units per primitive cell. The sym-metric unit cell shown in Fig. 2 contains two primitivecells. The structure is based on extended layers that runperpendicular to the b axis; parts of two layers are shownin Fig. 2. There is ionic and covalent bonding withineach layer, but only hydrogen bonding between layers.Within each multilevel, strong-bonding, layer there aretwo planes of aluminum atoms and four planes of oxygenatoms. Each aluminum is bonded to six oxygens in a dis-torted octahedron. The interior oxygens (those on thetwo inner oxygen planes) are each bonded to four alumi-nums in a highly distorted tetrahedron. The outer oxy-gens (actually hydroxyls, since each carries a hydrogen)are each bonded to only two aluminums.

An appropriate reference for comparison is o.-A1203(corundum, also called sapphire or alumina). In thethree dimensional (3D)-network structure of a-A1203,each aluminum is coordinated to six oxygens and eachoxygen to four aluminums. The average coordination is4.8, compared to an average Al-0 coordination of 4.0within a boehmite layer. For a-alumina, Raman' and in-frared' studies place the optical phonons in the rangefrom 380 to 750 cm '. For boehmite, the lower coordi-nation is expected to extend the Al-0 optical-phonon re-gime to lower frequencies, and we find this to be the case.

The strongest Raman line we observe in y-A10(OH)occurs at 363 cm '. Its strength, as well as the observa-tion (discussed below) that its depolarization ratio is zero,marks it as a fully symmetric mode ( A; there are threeof this symmetry), in which all aluminums and oxygensmove parallel to the b axis. The six vertical arrows inFig. 2 show one possible 3 -symmetry vibrational eigen-vector (we can treat each hydroxyl as a rigid entity, for alow-frequency mode, so only six arrows are needed)

FIG. 2. The crystal structure of boehmite y-A10(OH). This0

is a perspective view from a distance of 50 A, looking along adirection 15' oIt' the a axis in the ac plane (see Ref. 12). The cellshown is a symmetric unit cell, which contains two primitivecells. Oxygens are represented by the large spheres, aluminums

by the small spheres, and hydrogens by the solid dots. The sixarrows, attached to the aluminums and oxygens within oneprimitive cell, represent a vibrational eigenvector having the full

crystal symmetry.

RAMAN STUDIES OF SOL-GEL ALUMINA 15 629

363 boehmite

Alo(OH)677

CDC:

CD

G$OU)

496

200 300 400 500 600 700 800

Raman Shift (cm ')

FIG. 3. The Raman spectrum of crystalline boehmite in theAl-O regime. Scattered light intensity is shown plotted againstphoton-energy downshift, in wave-number units, relative to the

laser line (the 488.0 nrn line, for this case).

4000

M

0O

3000(0

(0

CDC:L(D 2000(QO(6

G5

ECC

k

x component

depolarization

xy component

340 360

Raman Shift (cm ')380

TABLE I. Raman peak positions for crystalline boehmite,Alo(OH).

Frequency(cm ')

230259270341363.3454496638677733

30793219

Linewidth'(cm ')

3.22.62.43.23.07.36.3

10.912.49.5

4737

Relativeintensity

WeakWeakWeak

MediumVery strong

MediumStrongWeak

MediumWeak

MediumMedium

'The linewidth means full width at half maximum (FWHM),and is corrected for the instrumental broadening via(FWHM), b, =(FWHM)„,„,&+(instrumental spectral slit width) .

variations of this feature, as observed for the gel samples,will play an important role in our analysis.

Figure 4 shows the polarization dependence of the 363cm line. (For this nanocrystalline sample, the line isdownshifted to 359 cm '.) These spectra are for a hydro-gel, peptized with HNO3 (R =0.07), and aged in the solphase for 48 h. These clear hydrogels are convenient fordepolarization studies because of their low scatteringbackground and the random orientation of the crystal-lites. The clean diagonal polarization exhibited in Fig. 4supports the fully symmetric 3 -symmetry interpretationfor the 363 cm ' Raman band.

In comparing our results to earlier reports on boeh-mite' ', we have discovered that considerable con-fusion has existed in terms of Raman lines attributed toboehmite and boehmite gels. Specifically, we have foundthat several high-frequency lines, which have masquerad-ed in the literature as boehmite modes, are actually inter-nal modes of the NO3 nitrate ion. Figure 5 compares

FIG. 4. The polarization dependence of the 363 cm ' line forboehmite. (For this nanocrystalline sample, the line is down-

shifted to 359 cm '. ) These spectra (obtained with 514.5 nm

excitation) are for a hydroge1, peptized with HNO3 (8 =0.07)and aged in the sol phase for 48 h. This clean depolarizationbehavior supports the fully symmetric Ag assignment for this

band.

spectra that we have observed for boehmite crystals(lower panel), an alumina hydrogel or "boehmite gel"(middle panel), and nitric acid (upper panel). The gel wasprepared as in Fig. 1, using HN03 as the peptizing acid.Note the four Raman bands indicated by the vertical

x15

H20

NO3

CD

CO

(QOV)

G5

E(Q

CC

L L~~gel

boehmitecrystal

500 1000 1500 2000

Raman Shift (cm ')FIG. 5. Comparison of the Raman spectra of (a) nitric acid,

(b) an alumina hydrogel, and (c) crystalline boehmite. (Thesespectra, as well as those shown in the following 6gures, were ob-tained with 514.5 nm excitation. ) The spectrum shown in (a)was obtained for a 0.1N solution of HNO3. The spectrumshown in (b) was obtained for a gel prepared using nitric acid in

the peptization step (at a HNO3. ASB molar ratio R of 0.14) and

a sol aging time of 48 h. The four NO3 bands marked by verti-

cal lines are seen to be present in the gel spectrum; thesenitrate-ion lines have been misinterpreted (as boehmite lines, oras lines due to organic impurities) in some previous studies ofsol-gel alumina.

15 630 C. J. DOSS AND R. ZALLEN

lines (at 720, 1049, 1348, and 1414 cm ' ), especially thestrong, sharp line at 1049 cm '. All of these are presentin both the HNO3 and the gel spectra, and all are absentin the spectrum of boehmite. These four bands corre-spond to NO3 vibrations. ' ' The symmetric-stretchnitrate line at 1049 cm ' has been erroneously assignedto boehmite, and the asymmetric stretching line at 720cm ' has been erroneously assigned to boehmite gels.Also, the 1348 and 1414 cm ' lines have been misas-signed as organic features in the boehmite gels. All boeh-mite gel spectra that have been reported to exhibit theselines were obtained on samples that were either peptizedwith HNO3 or involved Al(NO3)3 in their synthesis. Theattribution of extraneous lines to the Raman spectrum ofboehmite has prevented a correct group-theoretical inter-pretation of the zone-center vibrations in this crystal; weplan to rectify this in a future paper.

A comparison of the spectra in the lower two panels ofFig. 5 confirms that boehmite A10(OH) is, indeed, theprimary component (other than water) in the hydrogel.The 363 cm ' signature of the crystal is prominent in thege1, but, as we shal1 see, this line is actually broadenedand shifted with respect to the crystal line.

V3

(D

C:

CAC:L

(D

cU

CO

G5

EG$

CC

peakiposition

330 340 350 360 370

Ralnan Shift (cln ')

380

I I I Ii

f I I Ii

I I I Ii

I I I I/

I I I I

FIG. 6. Detailed view of the boehmitelike band in an alumi-na hydrogel, illustrating the line-shape parameters used in ouranalysis. The curve is a fit to data obtained for a gel synthesizedusing HC1 peptization (acid:ASB molar ratio R of 0.07) and a t,of 24 h. The linewidth indicated here is the experimentallinewidth; it is a convolution of the actual linewidth and the in-strumental spectral slit width. The linewidth values plotted inFigs. 8 and 9 correspond to the (deconvoluted) actual linewidth.

IV. RAMAN SCATTERING FROM ALUMINA GELS:SYSTEMATIC CHANGES DURING SOL-GEL

PROCESSING

In the synthesis procedure for sol-gel alumina that isoutlined in Fig. 1, the primary processing variable of in-terest in this study is the sol aging time t, . Secondarily,we have also studied the etfect of varying (or omitting)the acid peptization step. Although ge1 samples pro-duced after diA'erent times t, all exhibit Raman spectrawhich, in overall appearance, resemble that shown in Fig.5(b), we have observed definite systematic changes as t, isvaried. As noted in Sec. II, after gelation is achieved byconcentrating and cooling the aged sol, no furtherstructural changes are observed.

The gel Raman features that we have followed, as afunction of sol aging time, are the position, linewidth,and asymmetry of the prominent boehmite band near 360cm ~ Figure 6 shows a detailed view of this band, ob-served for a sample corresponding to t, =24 h. The ex-perimental linewidth I,„,(I,„,=lw+rw, where lw andrw are the left and right half-widths) is the observed fullwidth at half maximum. The asymmetry is defined as1w/rw. The experimental linewidth is a convolution ofthe actual linewidth I and the instrumental broadeningI,„„,I,„,=(I ~+I;„„)'~;in the following analysis wepresent the (deconvoluted) results for the estimated actu-al linewidth I .

Figure 7 shows the 360 cm ' band for a series of sam-ples with t, values varying from 1 to 72 h. In panels(a) —(d), the experimental data are shown as solid dots,while the lines are fitted curves useful for comparing andanalyzing the data. (The curves are eight-parameter fits,using two Cxaussians and a linear base line. The fittedcurves were used to determine the peak positions andlinewidths. ) Panel (e) of Fig. 7 shows a superposition ofthe curves; like panels (a) —(d), it demonstrates that the

~ ~ ~

~ ~~Q~~

hour~ w

~ ~ %%% +FAN )

urs

hours

hours

peak shifts and narrowswith tin hours

330 340 350 360 370Raman Shift (cm')

380

FIG. 7. The inhuence of sol aging time on the position andline shape of the boehmitelike band seen in the resulting hydro-gels. The solid dots in panels (a) —(d) are the experimental data;the curves are fits. Panel (e) shows a superposition of the fittedcurves of the top four panels, for t, values of 1, 8, 24, and 72 h inorder to more clearly reveal the narrowing and upshifting withincreasing t, . This series of hydrogels was prepared from a solthat had been peptized with HC1 (with R =0.07).

RAMAN STUDIES OF SOL-GEL ALUMINA 15 631

band upshifts and narrows with increasing sol aging timeat 95 C.

Figure 8, which represents our principal findings forthe inhuence of preparation procedure on the boehmiteband position and line shape in the sol-gel alumina ma-terials, summarizes results obtained from about 70 sam-ples, via spectra such as those in Fig. 7. Figure 8 presentsthe observed correlation between peak position andlinewidth. The main family of materials, represented inFig. 8 by the solid triangles, are the hydrogels preparedby the process of Fig. 1, with the peptization step carriedout with nitric or hydrochloric acid (with acid:ASB ratiosof 0.07 and 0.14). In addition, several other types of sam-ples were investigated and have been included in Fig. 8.Boehmite itself [via more detailed data than shown inFig. 5(c)] is included as the solid square labeled "bulkcrystal" at the lower left of Fig. 8. This corresponds tothe smallest linewidth and highest peak position for the360 cm ' band; it represents the limiting values.

The central result displayed by the extensive data con-tained in Fig. 8 is the position versus linewidth correla-tion represented by the solid straight line. This line is afit to the gel data (the solid triangles, about 50 samples) inthe upper part of the figure; the bulk-crystal point at thelower left (not included in the fit) lies close to the extrapo-lation of this line. Within this hydrogel data set, the gelsrepresented by points furthest to the upper right corre-spond to the shortest sol aging times. With increasing gelsample t„ the points track down and to the left, systemat-ically heading in the direction of the bulk crystal point.

A second result is indicated by the dashed line in Fig.8. This is a linear fit to the data set (the open squares)corresponding to the unpeptized precipitates prepared byomitting the acid addition step in Fig. 1. This family ofmaterials also exhibits an approximately linear position-linewidth correlation, but one that is slightly shifted withrespect to the line describing the correlation for the gelsamples.

Results for a few other types of sol-gel alumina samplesare included in Fig. 8. The xerogels (represented by theopen triangles) were made by drying selected hydrogelsamples (indicated by the tie lines) at 110 C. The 130 Cpowder precipitate was an unpeptized sample held at130'C for five days in a closed vessel. For all of thesesamples, the position-linewidth parameters define pointswhich lie in or near the narrow band marked by the twolines in Fig. 8. Interpretation of the correlations revealedby these data is discussed in the next section.

V. NANOCRYSTALLINE BOEHMITE

The points at the top right of Fig. 8, for which theboehmite-like band is substantially broadened and down-shifted relative to the bulk-crystal band, correspond toalumina gels prepared using the shortest sol times(t„t2, . . . ) in the procedure of Fig. 1. For longer reac-tion times, the band exhibited by the resulting gels is seento narrow and upshift; the spectral parameters tend to-ward those of the bulk crystal. This systematic variationin the Raman spectrum evidently probes a systematicvariation in the structure and/or the composition of thegel, as a function of sol aging time.

The main candidate for a stoichiometric explanation ofthe observed Raman changes involves the possible incor-poration of excess water within the boehmite phase. In-tercalation of water into the interlayer (hydrogen bond-ing) regions in boehmite has been suggested in the litera-ture, ' but clear evidence for this has not been presented.We consider a variation in interlayer water content to bean unlikely explanation of the results of Fig. 8. The bulkcrystal is the end point corresponding to very long agingtimes, which seems inconsistent with an excess water ex-planation.

It has also been suggested that bond-length and bond-angle variations might account for the increasedlinewidth in the gel. This idea implicitly appeals to an

356

358

0~~~~o 360—CL

05CDCL

cdE 362-C5

CC

10

20

(nm) FIG. 8. The correlation between peak posi-tion and linewidth for the boehmitelike Ramanband observed in sol-gel-alumina materials.Seventy samples are represented in this figure,over 50 of these being Yoldas-process hydro-gels synthesized using a range of sol agingtimes. The other samples are described in thetext.

364

linewidth

10

(cm ')

15

15 632 C. J. DOSS AND R. ZALLEN 48

analogy with silica gel, which is closely related to amor-phous SiOz, a covalently bonded, low coordination num-ber, continuous random network, bulk glass that ischaracterized by a broad distribution of Si-0-Si bond an-gles. But the analogy between alumina gel and silica gelis wrong for many reasons. The alumina system is pri-marily ionic, not covalent. There is no bulk aluminaglass analogous to silica glass. Coordination numbers incrystalline Alz03 are significantly higher than in SiOz. sixfor Al and four for 0 in A1203 versus four for Si and twofor 0 in SiOz. Silica glass is a classic Zachariasen struc-ture constructed of corner-sharing Si04 tetrahedra; thebridging oxygens provide the fIexibility needed to formthe random network. A1203, A10(OH), and Al(OH)3 areall based on edge sharin-g A106 octahedra (nearby alumi-num atoms are joined by two oxygen atoms); the geome-trical constraints enforce crystalline structures. Thus,alumina gel is predominantly microcrystalline or nano-crystalline rather than amorphous, and there is littlescope for bond-length or bond-angle variation.

The observed correlation between Raman linewidthand peak position, exhibited in Fig. 8 for the boehmite-like signature characteristic of the family of materialsknown as sol-gel alumina, bears a strong resemblance to awell-documented phenomenon observed in nanocrystal-line semiconductors: the finite-size effect of nanocrystal-linity (microcrystal dimensions of order 10 nm) on Ra-man position and line shape. The gist of the mecha-nism is as follows. For a nanocrystal of size L, the strict"infinite-crystal" k space selection rule is replaced by arelaxed version characterized by a k space uncertainty oforder (1/L). Here, k is the phonon wave vector. Insteadof only the k =0 mode of a phonon branch being Ramanactive (for a Raman-allowed mode symmetry), now allmodes of the phonon branch within a range of k values[roughly from k =0 up to k =(1/L)) are Raman active.Because the phonon dispersion curve U (k) is not flat, theobserved Raman band shifts and broadens. The smalleris L, the larger is the shift and broadening of the Ramanband. The x-ray results of the following section show, infact, that in our materials, L decreases in the decreasingtime direction (the direction of increasing shift and in-creasing broadening) in Fig. 8. The L scale along theright side of Fig. 8 is based on the x-ray-derived averagecrystallite sizes discussed in Sec. VI.

Both the clear correlation in Fig. 8 between peak-position downshift and linewidth broadening, as well asthe magnitude of the measured changes in these quanti-ties, are similar to what is observed in nanocrystallinesemiconductors. A second conspicuous feature of Fig. 8is the small but real offset of the line representing the pre-cipitate samples relative to the line representing the mainfamily of samples, the hydrogels. Before discussing thisinteresting effect, we first present, in Fig. 9, additionalsets of results for the line-shape changes.

Figure 9 shows the correlation between linewidth andline-shape asymmetry (upper panel of the figure) and be-tween peak position and asymmetry (lower panel). Herethe line-shape asymmetry information is contained in theratio (lw/rw), where lw and rw are the left and righthalf-widths illustrated in Fig. 6. For a symmetric band,

5 I \ I II

I I I II

I I

a

E10

E 358

O

V)OCL

G$Q)CL

360

362

3641 1.1 1.2

Is

atesbulk crystal ~c

1.3 1.4

asymmetry (Iw/rw)

FIG. 9. Correlations between the asymmetry of the boeh-mitelike band line shape and the position and linewidth of theband, for the samples of Fig. 8.

this ratio is unity; this is close to the case observed forbulk boehmite. The most asymmetric line shapes are ob-served for the short-time gels, for which the observed ra-tio is about 1.3. The asymmetry is small and thus thescatter is substantial, but Fig. 9 does show that linewidth,peak position, and asymmetry are mutually correlated.

In Fig. 8, there is an offset between the lines describingthe two large populations of samples: hydrogels and pre-cipitates. In Fig. 9(a), data from both populations overlapand no such separation or offset is evident. But in Fig.9(b), we can again see a definite separation, as in Fig. 8.Our interpretation of these observations, in terms ofnanocrystal-size arguments, is given below.

For a single nanocrystal diameter L, the finite-sizemodel yields a single value for each of the three Ramanband characteristics: peak position, linewidth, and line-shape asymmetry. For a distribution of nanocrystal sizeshaving a peak at L and a spread schematically denoted by5L, the Raman peak position is determined by L but thelinewidth and asymmetry are increased over the single L(monodisperse) value by contributions arising from theparticle-size spread 5L. %'e interpret the hydrogel-precipitate Raman band offset, discernible in Figs. 8 and9(b) but absent in 9(a), as evidence that 6L is larger forthe nanocrystal distribution in the precipitates than forthe nanocrystal distribution in the hydrogels. In otherwords, the precipitates exhibit more inhomogeneousbroadening than the hydrogels. A comparison of hydro-gel and precipitate spectra, for bands with similar peakpositions, reveals that the main difference is that the lefthalf-width (lw of Fig. 6) is larger for the precipitates.

48 RAMAN STUDIES OF SOL-GEL ALUMINA 1S 633

This accounts for the offset between the two populationsseen in Figs. 8 and 9(b), as well as the absence of an offsetin Fig. 9(a) (since an increase in lw increases bothlinewidth and asymmetry, approximately preserving thelinewidth-asymmetry correlation).

An increased low-frequency wing (larger lw) for theRaman band of the precipitates implies a larger spread inthe particle-size distribution on the small-particle side. Apossible explanation for an increase in the number ofsmall nanocrystals is the 110'C drying step experiencedby the precipitates (but not, of course, the hydrogels).During this step, the concentration of aluminum-containing solute species increases drastically, and atsome point it exceeds the critical supersaturation for nu-cleation. The resulting additional burst of nucleationgives rise to a population of small nanocrystals, which isresponsible for the increased dispersion on the small-Lside of the particle-size distribution.

To test this idea, we subjected three of the hydrogels tothe same 110'C drying step experienced by the precipi-tates. The results are included in Fig. 8 as the three pairsof triangles connected by tie lines. For each pair ofpoints, the filled triangle corresponds to the initial hydro-gel and the open triangle corresponds to the xerogel pro-duced by drying that hydrogel at 110 C. All three tielines move the initial point to the right, to largerlinewidth. Two of the xerogel points are seen to lie nearthe line describing the precipitates, while the smallest-Lcase has moved part way in the same general direction.These results are consistent with the picture of increasedsmall-particle dispersion caused by a late, drying-inducedburst of nucleation.

The open circle in Fig. 8 is for a precipitate which,directly after formation by hydrolysis and condensationin the 1:110ASB/water liquid, was transferred (precipi-tate plus liquid) to a closed pressure vessel and held at130 C for five days. It was then removed and dried at110 C, leaving a white powder. This sample can beviewed as an exercise in boehmite crystal growth. In Fig.8, its position is the closest one to the bulk-crystal point.It is also worth noting that it lies near the main line rath-er than the offset line, showing that the offset is not asso-ciated with the absence of acid (since this sample wasacid-free).

The nanocrystal particles in contact with the sol (thenanocrystals in the colloidal sol, for the liquids that pre-cede the hydrogels, and the nanocrystals forming the pre-cipitate for the unpeptized stirred suspensions) arepresumed to grow in size by the dissolution and reprecip-itation of an amorphous Al-containing component, orpossibly an assembly of clusters, that is formed in the ini-tial hydrolysis and condensation reactions between ASBand water. Such a source of alumina can account for thegrowth of the boehmite signal that we see. In the case ofthe 130 C precipitate discussed above, it appears that thesource has been depleted during the five day period ofcrystal growth, since we see no spectral evidence of a finalburst of nucleation caused by the drying step.

The crystalline character of all of these sol-gel materi-als (hydrogels, xerogels, precipitates) is attested to by theunambiguous signature of crystalline boehmite in the Ra-

X-ray-diffraction experiments were performed onseveral of the materials for which Raman results weregiven in Fig. 8. Measurements were done on samples inpowder form, as described in Sec. II D. The x-raylinewidth results were used to estimate average crystallitesizes by means of the Scherrer equation, Eq. (1).

Figure 10 shows the x-ray diffraction data. The peak

~ ~(6

0)~~CD

LQ)

G5O0)

05

DL.

CUC)

20 30 40 50

28 (degrees)FIG. 10. X-ray-diffraction data taken on powder samples, us-

0ing Cu Ea radiation (1.54 A wavelength). The very sharp peaksin trace a, obtained with micron-size crystals, are consistentwith the calculated diffraction spectrum of bulk crystallineboehmite. The peaks in traces b—g show progressively broaderlinewidths, indicating progressively smaller crystallite sizes.Crystallite sizes were calculated, using the Scherrer formula[Eq. (1)j, from the widths of the (021) and (131) peaks. Thebulk-crystal sample corresponding to trace a is represented by asolid square in the Raman data of Figs 8, 9, and 11. Trace b isthe 130'C precipitate described in the text and is represented byan open circle in Figs. 8 and 11. Traces c—e are 48, 8, and 1 hprecipitates, respectively (open squares in Figs. 8, 9, and 11).Trace f is for a 48 h gel peptized with nitric acid at R =0.07(solid triangle in Figs. 8, 9, and 11). Trace g is for a 1 h xerogel(open triangle in the upper right-hand corner of Figs. 8 and 11).

man spectra. The six strongest lines of the crystal (allthose listed as medium or stronger in Table I, except forthe 733 cm ' line) all appear in the gel spectra. In addi-tion, transmission electron diffraction on gel films(prepared by allowing an aged sol to dry by evaporationon a sample grid) exhibited many sharp diffraction rings.X-ray-diffraction data, described in the next section,confirm the crystallinity and provide a valuable calibra-tion of microcrystallite size.

VI. X-RAY-DIFFRACTION DETERMINATION OFNANOCRYSTAL SIXES

15 634 C. J. DOSS AND R. ZALLEN

assignments are given with respect to the y-A10(OH) or-thorhombic Cmcm crystal structure. " At the bottom(trace a) is the result obtained for a powder composed ofmicron-size crystals; this trace exhibits very narrowlinewidths (about 0.1') and is the pattern expected forbulk crystalline boehmite. The traces b —g show progres-sively broader linewidths (up to a few degrees), indicatingprogressively smaller crystallite sizes. The low-angle(020) band becomes asymmetric and shifts to higher dvalues as the linewidths increase, while the (021) and(131) bands remain symmetric and do not shift withlinewidth broadening. The (021) and (131) peaks wereused for crystallite size determination by the Scherrermethod. They gave consistently similar results, as seen inFig. 11.

Figure 11 presents the correlation between the crystal-lite size, as determined by x-ray diffraction, and the Ra-man peak position and linewidth (of the 360 cm ' band)observed for the same sample. The symbols used in thisfigure correspond to the same convention used in Fig. 8.For each Raman determination (vertical location) thereare two x-ray determinations (horizontal locations), cor-responding to the (021) and (131) diffraction peaks. Therange of crystallite sizes is seen to extend down to about 3nm (justifying our use of the terms nanocrystal and nano-crystallinity).

The top x-axis scale in Fig. 11 gives the crystallite sizeL; the bottom x-axis scale gives L '. The scale along theupper right side gives the downshift Av of the Raman

nanocrystal size, L (nm)20 10 7 5 4

peak position, relative to the peak position for the infinitecrystal. The two straight lines are least-squares linear-in-L ' fits to the data:

b, u =k, (1/L)

I =k~(1/L)+I o, (3)

VII. KINETICS OF NANOCRYSTAI. GROWTH

Figure 12 displays the results of individual sample-preparation runs, showing the dependence of the boeh-mite signature Raman band peak position of the hydrogelon the sol aging time (t, in Fig. 1). Results are shown for

where I is the linewidth, and kI, kz, and 10 (21.5cm ' nm, 39.2 cm ' nm, and 2.7 cm ', respectively) aredetermined from the fit. These calibration relations, con-necting Raman peak-position and linewidth with crystal-lite size, were used to obtain the right-hand scale as inFig. 8. They are also used in the next section, to deter-mine L(t, ) growth kinetics.

A linear relationship between Rarnan peak positionand L, with a=1.0, has also been seen in graphiteand boron nitride. It is noteworthy that boehmite, likegraphite and BN, possesses a layered structure (as dis-cussed in Sec. III). In terms of a, which is an empiricalRaman-versus-size scaling exponent, these layered ma-terials behave differently than the covalent network semi-conductors silicon ' and gallium arsenide. ' Ouranalysis of the reported Raman results ' ' for the net-work semiconductors yields, for those materials, a scalingexponent o; of about 1.5.

356

E(3

358

360

362

E10

position

linewidth

6

~~peak

4

(cm ')2

15

— 10

360

358ED

356

c 36000

358cUCO

356 '

c5 9360—65

CC

358

356, I I I s I I i I

0 20 40 60 0 20 40 60 0 20 40 60 80

0.10 0.20 0.30 time (hours)

L (nm ')

FIG. 11. The correlation between the crystallite size (L), asdetermined by x-ray diffraction for a particular sample, and theRaman peak position and linewidth (of the 360 cm ' band) ob-served for the same sample. The samples and their correspond-ing symbols have been described in the caption of Fig. 10.

FIG. 12. The results of individual runs on the inhuence ofthe sol aging time on the Raman peak position of the boehmite-like band in the hydrogel. The acid used in the sol peptizationstep is indicated in each case, as well as the acid:ASB molar ra-tio R. One unpeptized example (R =0) is included in panel cfor comparison. The peak-position scale in this figure is reversedwith respect to the scales used in Figs. 8 and 9(b).

48 RAMAN STUDIES OF SOL-GEL ALUMINA 15 635

both of the acids used in the peptization step, HCl andHNO3, and for both acid concentrations used, R =0.07and R =0.14 (R is H:Al, the acid:ASB mole ratio). Alsoincluded (panel c) are results for the precipitates obtainedin an acid-free run. All of the runs were well representedby fits of the form

v(t) —v(0)=rt'i (4)

The curves shown in Fig. 12 correspond to Eq. (4), withthe best-fit parameters, v(0) and r, listed in Table II. Theprecipitate behavior clearly differs from the rest in itslarger v(0). For the gels, v(0) is smaller for the higheracid concentration, but there is no discernible differencebetween the use of HC1 and HNO3 as peptizing agent.The rate constant r increases with increasing acid concen-tration.

As the characteristic microcrystallite size increases,the finite-size-induced downshift of the Raman peak posi-tion is reduced, and the peak moves upward toward itsbulk-crystal value of 363 cm '. Using the results of thelast section, we can make this interpretation quantitativeby invoking the b,v(L) scaling relation of Eq. (2), whichconnects the left and right scales of Fig. 8. Each set ofv(t) data, corresponding to the set of points contained ina panel of Fig. 12, is thus transformed to a set of L(t)data, with I the crystallite size. The results obtained forpanels c and e of Fig. 12 are presented in Fig. 13.

All of the hydrogels represented in Fig. 12 yield L(t)results similar to the HC1 R =0.14 data (panel e of Fig.12, lower part of Fig. 13) shown in the lower part of Fig.13. The precipitate run (panel c of Fig. 12, upper part ofFig. 13) is distinctly different. The curves shown in Fig.13 are fits of the form,

[L,(r ) ]'—[L,(0)]'=S,r,where L3(0) and S3 are the fit parameters. This formwas found to adequately describe all of the L (t) results;the values obtained for L 3(0) and S3 are given in the lasttwo columns of Table II. Also included in Table II areresults to fits of the form,

[L,(t) ]'—[L,(0)]'=S,(t),that also describes the experimental results, within thestatistical error bars shown.

(D

C5

4C3

20 40 60 80

The time dependence exhibited by L2 3(t) is similar tothat found in various theories of crystal growth,which come under the general heading of Ostwald ripen-ing. Ostwald ripening (coarsening, coalescence, or justripening) is basically a zero-sum game in which largerparticles grow at the expense of smaller particles; for ourpurposes, the larger particles are nanocrystalline boeh-mite, and the smaller particles are amorphous or incom-pletely crystallized A1O(OH). Our results are quite con-sistent with the often invoked cubic-law behavior [Eq. (5)]corresponding to the fitted curves in Fig. 13, but our dataare not accurate enough to rule out parabolic-lawbehavior [Eq. (6)].

The unpeptized precipitate sample included in Figs. 12and 13 and Table II is characterized by a larger initialcrystallite size [L (0)] and by a faster growth rate (s)than the peptized sols. Acid peptization limits early crys-tal growth. Overall, both acids used for preparation ofthe sols yield quite similar results. No difference is ob-

time (hours)

FIG. 13. Nanocrystal-size growth kinetics for the runs ofFigs. 12(c) and 12(e), estimated from the Raman and x-ray databy the method described in the text.

TABLE II. Parameters obtained from the rate-equation fits [Eqs. (4)—(6) in the text] to the data ofFigs. 12 and 13.

Panelin Fig. 12 Acid

Acid concentration U(0) r L2(0) s2 L3(0) 3(H:Al) (cm ') (cm ' h '

) (nm) (nm h ') (nm) (nm' h ')

HC1HClnoneHC1HC1

HNO3HNO3HNO3HNO3

R =0.07R =0.07R =0.0R =0.14R =0.14R =0.14R =0.07R =0.07R =0.07

356.4356.3358.1

355.7355.9355.8356.1

356.2356.1

0.320.310.330.440.390.380.300.250.33

3.33.24.53.03.1

3.03.1

3.23.2

0.220.190.700.270.240.210.180.140.20

3.33.24.4'3.03 0'3.0F 1F 13.1

1.31.163'1.51.5'1.21.00.71.1

'The curves that correspond to these parameters are shown in Fig. 13.

15 636 C. J. DOSS AND R. ZALLEN

served between HC1 and HNO3 in terms of the L(0)values obtained, but increasing the the acid concentrationfrom R =0.07 to R =0.14 does reduce L (0) slightly, aneff'ect which may reffect faster peptization (and reducedopportunity for early growth) at the higher concentra-tion. The subsequent growth rate seems to be slightlylarger with HC1 than with HN03 and, for both acids,growth occurs somewhat more quickly at the higher acidconcentration. But in all cases the growth is very slow,so that the gelation step yields hydrogels of nanocrystal-line boehmite.

VIII. SUMMARY

have been interpreted in terms of nanocrystallinity-induced finite-size effects associated with the slow growthof the y-A10(OH) nanocrystals during the sol phase.X-ray-diffraction experiments were used to determinenanocrystal sizes (as small as 3 nm for gels prepared fromfresh sols) and to estimate growth kinetics from the Ra-man line-shape results. The Raman peak-position shift isproportional to I. , where I, is the average nanocrystalsize and a is a Raman-versus-size scaling exponent. ForA10(OH) we find a to be 1.0, close to scaling-exponentvalues reported for graphite and BN and different fromthe value of 1.5 which describes the reported behavior ofSi and GaAs.

A systematic Raman-scattering investigation has beencarried out on sol-gel alumina prepared by the Yoldasprocess of Fig. 1, primarily as a function of the sol agingtime, but also, secondarily, as a function of the acid pep-tization step (including its omission). The predominantaluminum-containing component of the alumina hydro-gels and precipitates is nanocrystalline boehmite,y-A10(OH), and in the course of this work we haveclarified certain aspects of the boehmite Raman spec-trum.

A careful study was made of small but systematicchanges observed, as a function of the process parame-ters, in the line shape and position of the dominant boeh-mite band near 360 cm . The position-linewidth corre-lation established by the data in Fig. 8, as well as themore sUbtle line-shape asymmetry correlations of Fig. 9,

ACKNOW I.KDGMKNTS

This research was supported in part by Philip Morris,U.S.A. We are grateful to A. G. Kallianos and A. C. Lil-ly for their advice and encouragement in the initial phaseof this work. We are indebted to K. Wefers for providingthe crystalline boehmite sample and G. V. Gibbs formaking available to us the Bukowinski-Downs crystal-structure graphics program used in constructing Fig. 2.S. Joardar provided invaluable assistance in materialssynthesis techniques. We thank K. Wefers, N. Rivier,and A. L. Ritter for valuable discussions, S. McCartneyfor electron diffraction and TEM work, and T. N. Solbergand M. Ferguson for very valuable help with the x-ray-diffraction experiments.

iB. E. Yoldas, Am. Ceram. Soc. Bull. 54, 286 (1975).B.E. Yoldas, Am. Ceram. Soc. Bull. 54, 289 (1975).B.E. Yoldas, J. Mater. Sci. 10, 1856 (1975).

48. E. Yoldas, J. Appl. Chem. Biotechnol. 23, 803 (1973).5C. J. Brinker and G. W. Scherer, Sol-Gel Science (Academic,

Boston, 1990), p. 599.Temperature is an important variable in the Yoldas process. If

cool water is used, Al(OH)3 predominates and forms mixed-phase precipitates. A Rarnan-scattering investigation of thedistinction between the "high-temperature" sol-gel regime forthis system (temperatures close to 95 'C) and the "low-temperature" precipitate regime (temperatures close to 25 C)will be presented by us in a subsequent paper. The presentpaper is concerned with the high-temperature regime that isoptimum for the preparation of alumina gels.

7C. J. Brinker and G. W. Scherer, Sol-Gel Science (Academic,Boston, 1990), p. 371.

Our simple deconvolution is a standard approximation whichis strictly correct for Gaussians. A more sophisticatedanalysis for Lorentzians has been given by J. Gonzalez-Hernandez, G. H. Azarbayejani, R. Tsu, and F. H. Pollak,Appl. Phys. Lett. 47, 1350 (1985). Since, for most of our sarn-ples, r, pt is substantially greater than I;„„,the instrumentalcorrection is small and I is insensitive to the deconvolutionprocedure.

A. C. Pierre and D. R. Uhlmann, in Better Ceramics ThroughChemistry, edited by C. J. Brinker, D. E. Clark, and D. R. Ul-rich (Elsevier, New York, 1984), p. 119.

~oB. E. Warren, X ray Di+raction (Addison-Wesl-ey, Reading,MA, 1969), pp. 253 and 258; H. P. Klug and L. E. Alexander,

X ray Diffraction Proc-edures for Polycrystalline and Amorphous Materials (Wiley, New York, 1973), pp. 687 and 635.C. E. Corbato, R. T. Tettenhorst, and G. G. Christoph, ClaysClay Miner. 33, 71 (1985).

2Figure 2 was constructed using the atom positions of Corbato,Tettenhorst, and Christoph (Ref. 11), with the aid of acrystal-structure graphics program written by M. Bukowinskiand R. Downs.S. P. S. Porto and R. S. Krishnan, J. Chem. Phys. 47, 1009(1967).

~4A. S. Barker, Phys. Rev. 132, 1474 (1963).This sample, prepared at Alcoa Laboratories, was made by au-toclaving aluminum metal in excess water. X-ray diffraction,performed at NASA Langley, con6rmed that the materialwas well-crystallized boehmite.A. B. Kiss, G. Keresztury, and L. Farkas, Spectrochim. Acta,Part A 36, 653 (1980). Peak positions reported in this paperagree well with our results of Table I, with one notable excep-tion (in addition to the NO3 modes discussed in the text}.Kiss, Keresztury, and Farkas place the dominant line at 369cm ' instead of 363 cm '. Since our result was repeatedlychecked and is also consistent with the asymptotic limit ofthe family of gel samples, the 363 cm ' value is the correctone.G. Mariotto, E. Cazzanelli, G. Carturan, R. Di Maggio, andP. Scardi, J. Solid State Chem. 86, 263 (1990).T. Assih, A. Ayral, M. Abenoza, and J. Phalippou, J. Mater.Sci. 23, 3326 (1988).D. E. Irish and A. R. Davis, Can. J. Chem. 46, 943 (1968); D.E. Irish and G. E. Walrafen, J. Chem. Phys. 46, 378 (1967);

RAMAN STUDIES OF SOL-GEL ALUMINA 15 637

D. E. Irish, in Ionic Interactions from Dilute Soiutions toFused Salts, edited by S. Petrucci (Academic, New York,1971),p. 187.

P. Scardi, J. Solid State Chem. 86, 263 (1990).2'K. Wefers and C. Misra, Oxides and Hydroxides of Alumi

num, ALCOA (Alum. Co. Am. ) Res. Lab, Tech. Pap 19, re-vised (ALCOA Laboratories, ALCOA Center, 1987).

22R. Zallen, The Physics of Amorphous Solids iWiley, NewYork, 1983).K. K. Tiong, P. M. Amirtharaj, F. H. Pollak, and D. E.Aspnes, Appl. Phys. Lett. 44, 122 {1984);M. Holtz, R. Zallen,O. Brafman, and S. Matteson, Phys. Rev. B 37, 4609 (1988).I. H. Campbell and P. M. Fauchet, Solid State Commun. 58,739 {1986).

25R. Zallen, J. Non-Cryst. Solids 141, 227 (1992).6T. Kanata, H. Murai, and K. Kubota, J. Appl. Phys. 61, 969

(1987)~

27S. J. Yu, H. Asahi, S. Emura, H. Sumida, S. Gonda, and J.Tanoue, J. Appl. Phys. 66, 856 (1989).K. Nakamura, M. Fujitsuka, and M. Kitajima, Phys. Rev. B41, 12 260 (1990).R. J. Nemanich, S. A. Solin, and R. M. Martin, Phys. Rev. B23, 6348 (1981).

G. G. Christoph, C. E. Corbato, D. A. Hofmann, and R. T.Tettenhorst, Clays Clay Miner. 27, 81 (1979).H. Richter, Z. P. Wang, and L. Ley, Solid State Commun. 39,625 {1981).Z. Iqbal and S. Veprek, J. Phys. C 15, 377 (1982).

3M. Gargouri, B. Prevot, and C. Schwab, J. Appl. Phys. 62,3902 (1987).

4It is provocative that, for the cases examined, a is about equalto d/2, where d is the network dimensionality (e.g. , two forgraphite, three for silicon); see R. Zallen, The Physics ofAmorphous Solids (Wiley, New York, 1983), p. 89.

~I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35(1961); C. Wagner, Z. Elektrochem. 65, 581 (1961); W. W.Mullins, J. Appl. Phys. 59, 1341 (1986); N. Rivier, in FiniteSystems: From Clusters to Crystals, edited by P. Jena, S. N.Khanna, and B. K. Rao (Kluwer, Dordrecht, 1992), p. 189; P.W. Voorhees and M. E. Glicksman, Metall. Trans. 15A, 1081(1984).

A simple argument for the parabolic law is available forgrowth that is driven by crystallite curvature; growth rate(d/dt)L' is proportional to surface area (L') and to curva-ture (1/L), so that L(dL/dt) is constant and L increaseslinearly with time.