Modeling Particle Size and Clumping Effects in the IR Absorbance Spectra of Dilute Powders

CARLOS PECHARROMAN* and JUAN E. IGLESIAS! /nstituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cient{jicas, Cantoblanco, 28049 Madrid, Spain

The calculation of the IR absorbance of a powdered crystalline, ionic material, diluted in an appropriate IR-transparent matrix in a known proportion, is in principle possible, provided that the size and shape distributions of the particles, as well as the optical pa­rameters of the bulk material, and those of the matrix, are known, and, in the case of the FT-IR technique, whether absolute calibra­tion is possible, particularly at high absorbance values. In practice, a close match between observed and calculated absorbance is hin­dered by several factors intrinsic to the sample that are difficult to quantify, such as the degree of crystallinity of the particles and the very common clumping and aggregation of these particles, which produce a serious problem in the determination of the real filling factor of the composite sample. Some remedies to these problems are proposed in this paper and are applied to several well-known materials.

Index Headings: Infrared; Absorbance calculations.

INTRODUCTION

Infrared spectroscopy of solids is a widespread tech­nique employed in polymer research, materials science, solid-state physics, and inorganic chemistry, among oth­ers. The customary method of obtaining the IR absorp­tion spectrum of a solid requires one to disperse small particles (typically 0.1-1% in volume for inorganic sub­stances) of the material in an IR-transparent matrix, such as an alkali halide, KRS-5, polyethylene, etc. For molec­ular samples, this procedure is amenable to give accurate values of the frequencies of the vibrational modes be­cause they appear as narrow absorption maxima. In in­organic solids the interpretation of the IR absorption spectra is encumbered, especially for ionic substances, because the absorption maxima become broad due to large transverse-to-longitudinal frequency splitting (w1

wL), so that they must be referred to as bands. Addition­ally, the position of the band maxima is a function of the external shape of the particles 1 and of the filling factor. 2

Several approximations have been developed to ex­plain the IR spectra of ionic substances. The first method, published by Frohlich, 1 is based on the calculation of the scattering cross section of a small particle. This formu­lation explains qualitatively the IR spectrum of very di­lute samples, since only scattering effects by non-inter­acting particles are taken into account. Better results were obtained by Genzel and Martin,2 who employed an old expression for the calculation of the average dielectric constant of a dilute system, first introduced by Maxwell­Gamett,3 in which dipolar interactions between particles are implicitly considered. Using this expression, they were able to calculate the IR absorption spectrum of

* Present address: Dept. of Physics, SUNY, Stony Brook, NY 11764. t Author to whom correspondence should be sent.

small spherical particles of MgO dispersed in air. More recently the Maxwell-Garnett theory (which we will refer to in what follows as MGT) has been extended to systems with particle shapes other than spherical, and calculations carried out with these approximations have reproduced in a semi-quantitative way the experimental absorbance spectra of different substances.4-8 Among these general­izations the most significant are the expression due to Fuchs,9 which can be applied to small cubic particles, and that of Hayashi et al., 10 developed for optically aniso­tropic ellipsoids.

In spite of this success, several problems still arise when MGT is employed to fit experimental spectra. First, due to aggregation problems, it is often necessary to use in computations values of the filling factor that are much greater than the true average value if the experimental absorbance maxima are to be matched in frequency; these high values make MGT lose its validity because the hy­pothesis of high dilution employed in its deduction is no longer fulfilled. Also, the experimental bands are always found to be wider than the calculated ones. Bohren and Huffman4 have shown that a continuous distribution of shapes can broaden the IR bands. However, wide bands have also been detected in samples with a very narrow shape distribution,D-8 and it has become customary to as­sign this effect to higher values of the damping constants of the vibrational modes of the microparticles relative to those of the single crystal. Theoretical studies on Mg0 11 •12 predict that the difference in damping constant should be not large, a factor of two being a worst-case estimation. This factor is not enough to justify the broad­ening observed in the IR absorption bands.

A comprehensive revision of this field has been written by Huffman 13 detailing all the problems found in the study of the optical properties of small particles. Al­though there are other factors to be considered, the main problems are the lack of agreement between experimental and calculated absolute absorbance near the Frohlich modes and the excessive width of the experimental ab­sorbance bands in comparison with the theoretical band­width. In fact, most of the experimental spectra published are reported with arbitrary absorbance units, and this has prevented validation of the commonly accepted hypoth­eses. In order to test systematically these hypotheses, we have performed some experiments to detect the variation of position and shape of the IR absorption bands with aggregation and degree of crystallinity. We have also in­troduced a new procedure to calculate the IR optical properties of a spectroscopic sample based on the Effec­tive Medium Theory, pioneered by Bruggeman 14 and developed by Polder and Van San ten 15 and Landauer. 16•17

Our contribution is a generalization from that of Lan-

Volume 50, Number 12, 1996 0003-7028/96/5012-1553$2.00/0 APPLIED SPECTROSCOPY 1553 Cl 1996 Society for Applied Spectroscopy

FIG. 1. Optical micrograph of small particles of a-Fe20 3 as they appear in a KBr pellet. It can be seen that these. small particles tend to lie on the external surface of large KBr crystals.

dauer, but introducing 18 the concept of "Percolation Threshold" in a natural way. This approximation, which implicitly includes multipolar terms of interaction be­tween inclusions, instead of only dipolar ones (as in MGT), has revealed itself as a useful tool to properly describe the IR properties of aggregated systems, for ma­terials of known optical properties 18·19 as well as for oth­ers whose properties can be thereby determined, as is the case for -y-Fe20 3 (maghemitef0 and transition alumina T)-Al20 3 •21 In this paper, we show how calculated spectra can be made to better fit, both qualitatively and quanti­tatively, the experimental data.

EXPERIMENTAL

Monodispersed colloidal particles of Al2Mg04 (spinel), a-Fe20 3 (hematite), Ti02 (rutile), and MgO were em­ployed. Both MgO and Al2Mg04 were synthesized by Spray Pyrolysis,22 and we also used an additional com­mercial sample of MgO composed by cubic particles. The iron oxide samples were obtained by a hydrolysis syn­thesis procedure,23- 25 and the Ti02 samples were obtained by an aerosol method.7 The composition, crystallinity, and size distribution of the microparticles have been char­acterized in previous articles.5- 7

For quantitative measurements, small quantities of these monodispersed samples were carefully weighed and mixed with KBr powder. The transmission spectra were measured in a 20 SXC FT-IR Nicolet spectrophotometer fitted with a beamsplitter of Ge on Csl, with a resolution of 2 cm- 1• For each spectra, 200 scans were made, and the interferograms were signal-averaged, phase-corrected and apodized with the Happ-Gcnzel function, prior to Fourier transformation. The transmission spectra of pure

KBr was used as a reference. The linearity properties of the Happ--Genzel function have not been investigated in detail,:): but apodization functions are known to introduce non-linearities which cause the absorbance to be under­estimated for high values of this magnitude. We estimate that our measured absorbances have an error on the order of 15% whenever A > 2.0 A.U.

PARTICLE AGGREGATION AND EFFECTIVE FILL FACTOR

The most popular preparation method for making IR absorbance measurements on powdered samples consists of dispersing a small amount of the powdered substance, at concentrations ranging between 0.1% and 1 %, in an alkali halide powder, typically KBr or any other IR-tran­sparent medium. The mixture is then pressed in a vacuum under an approximate pressure of 0.5 to 1 GPa. In this process, the alkali halide flows and the matrix crystallites grow under the pressure, producing a clear, self-support­ing disk where the microparticles of the sample are dis­persed.

Because of the high dilution of the sample, it has been customarily assumed in the past that the sample micro­particles were homogeneously dispersed into the matrix. Since there is evidence that in many occasions this is not the case, we have made KBr pressed disks of iron oxide (hematite) and have observed them by visible and SEM microscopy. Optical microscopy reveals the pellet as a mosaic of single crystals of KBr grown by pressure sin­tering (Fig. 1 ). On the surface of these large crystals, dark regions with swarms of a-Fe20 3 spherical particles are

:j: We are grateful to an anonymous reviewer for pointing this out to us.

FIG. 2. SEM micrograph of the surface of a pellet of a-Fe20 3 in a matrix of KBr; a high concentration cluster of a-Fe20 3 particles can be observed.

observed. It appears that the KBr crystals push out the sample particles as they grow, with the result that these tend to concentrate at the intergranular boundaries. In or­der to further verify that conclusion, a fragment of a sim­ilar pellet was observed by SEM. We studied the outside surface of the disk and, also, an internal section obtained by cracking the pellet apart. In the external surface of the pellet (Fig. 2) a conspicuous aggregate of particles of a-Fe10 3, resembling shot, can be seen. In Fig. 3 it can be seen that, even in a well-dispersed pellet, aggregation in small clusters is present all over the sample, and that large particle-free regions can be found in the inside of the pellet. A similar sample behavior was detected by McLachlan26 in graphite/polymer composites, in which an extremely low threshold for electrical conductivity had been measured.

An estimation of the effective filling factor for IR ab­sorbance can easily be made if it is supposed that the pellet is formed by relatively large, spherical, matrix sin­gle crystals, and that these are free of included particles. The inclusion particles are assumed to lie on the spherical surface of the matrix single crystals (Fig. 4 ). The effec­tive volume V,ff in which the particles are confined is then reduced to

(1)

where N is the number of matrix spheres contained in the pellet, r the radius of the inclusion particles, and R the radius of the matrix crystals, and where the assumption r ~ R -7 Rr2 , r3 << R2r has been made. Clearly,

(2)

If n is the number of inclusion particles in the sample, then the effective filling factor is given by

".iii·= ::;p;

FIG. 3. SEM micrograph of the inside of a KBr/a-Fe20 3 pellet. Small clusters and also isolated particles are apparent.

(3)

The theoretical filling factor, i.e., the volume ratio of sample particles in the disk, is given by

(4)

which through substitution of Eq. 2 gives the following:

FIG. 4. Idealized model consisting of large matrix particles and smaller inclusion particles packed together inside the pellets used for IR spec­troscopy.

APPLIED SPECTROSCOPY 1555

4 n-7Tr3

_3 = ~n _Rr)3

• j,h = 4 N-7TR3

3

(5)

Using this result in Eq. 3 we obtain an estimation of the effective filling factor,

R Jeff = 3rfrh- (6)

We see that the ratio Rlr gives the order of magnitude of the factor by which the effective filling factor is in­creased relative to the theoretical one. In a real case, the average size of the KBr spherical single crystal can be as small as 4 1-Lm (which is the minimum size of a KBr particle before sintering under pressure), but usually the average size is about 20-50 1-Lm after the crystals have grown. In most occasions, the sample particles are around 0.1 to 0.5 1-Lm in diameter, so the effective filling factor can be 3 to 200 times the value of the theoretical one. This means that an accurate estimate of the effective fill­ing factor of a substance diluted in an ordinary IR pellet is very difficult to make, as the effective concentration depends on the average size of the KBr crystals in the pellet, which in tum depends on the pressing process. Since the usual value of the theoretical filling factor in such pellets is, for very absorbing materials, about 0.001 (0.1% ), it should not be surprising to find samples in which the effective filling factor could be as high as Jeff = 0.01 to -0.1, i.e., ten to one hundred times the theo­retical value.

Equation 6 also indicates a way to enhance the dis­persion of the sample particles in a KBr pellets-if R can be made to decrease, the inclusions will intermix more thoroughly with the matrix particles. We have tried two different procedures to reduce the matrix particle size. The first one has been to subject the disk material (matrix plus sample) to intensive grinding prior to pressing, thus provoking a reduction of the average size of the KBr single crystals and shuffling all the particles in the pellet. The second procedure has consisted in successively ap­plying to the pellet cycles of pressure-sintering and grind­ing. We have checked that both procedures lower the ef­fective filling factor. In the case of magnetic particles (iron oxides, for example), we observe6 that the weak attractive forces between the free particles tend to make them form clusters under the microscope, and then, as seen in Figs. 1-3, this effect is compounded when the KBr crystals grow. For these kind of substances, the ef­fective filling factor in some parts of the pellet could be very high, even the same order of magnitude of matrix­free pressed powder (j = 0.2 to -0.5).

ESTIMATION OF ABSOLUTE ABSORBANCE

In order to deal with the large variations that the ef­fective filling factor can exhibit in a real case, we have designed a simple model pellet to reproduce the experi­mental spectra. We suppose that the particles are distrib­uted in the pellet in small regions, in such a way that the effective filling factor inside the ith region is constant, f,. Due to the fact that the pellet is macroscopically homo­geneous, we can redistribute the volume of these regions

lr

lr

FIG. 5. The total absorbance of a heterogeneous mixture can be con­ceptualized by assuming that the sample is formed by several homo­geneous slices, each having a different partial concentration of the ma­terial under study.

into thin slabs parallel to the disk surface. With such a scheme (Fig. 5), an IR beam crossing the sample is suc­cessively absorbed by regions with different filling fac­tors and transmitted through a transparency region cor­responding to pure matrix material, designated as the zer­oth region. The final absorbance is the sum of the partial absorbances of all the slabs.

A = ~ A,(j,, z) (7)

with j; the true filling factor of the ith slab whose thick­ness is Z;-

Volume conservation of the included particles requires that

(8)

where S and d are, respectively, the surface area of the base and the thickness of the disk, vp is the total volume of particles, and f is the theoretical filling factor for the whole sample. The value of Vpi• the volume of all parti­cles in the ith slab, is given by the following:

vp, = /; v, = /;z,S.

Summation over all slabs in Eq. 9 gives N

2: z,/; = df, 1=1

with _ c,Ve

Z, - /;S

(9)

(10)

The value of z, can be calculated if we know what is the relative proportion c, = VPJVP of sample which ap­pears with an aggregation state described by/;.

Then, the total absorbance is given by

41T" A = T -'7 Z;K(/;) (11)

where K(/;) represents the imaginary part of the refractive index of the ith slab.

As explained above, it is difficult to obtain a reliable value of K(/;) for a heterogeneous mixture when/; :::=: 0.1 if a low concentration theory is used. As we have shown before, 18 the limit of applicability of MGT is f ~ 0.1, which explains the poor results obtained with this ex­pression in the case of samples which are very aggre­gated. We have suggested instead, the use of a general­ization of the so-called effective medium theory, intro­ducing a new parameter to describe a heterogeneous sys­tem, namely the percolation threshold. For a mixture of optically isotropic spheroids (revolution ellipsoids) we have 18

(1 - /)((E) - Em) [(1 - Lm)~) + LmEm

+ 2Lm(E) + (~ - 2Lm)EJ

+ /((E) - e.p) [o - Lp)~E> + LPEP

+ 1 l = 0 (12) 2LP(E) + (1 - 2LP)EP

where E represents the dielectric constant, and where sub­script p stands for particle and m stands for matrix.

We have found that the percolation threshold f.: is re­lated to the depolarization factor of the matrix Lm through

with

and

H f.:=H+G'

[ 2 1 l H= ---+-1 - Lm 2Lm

G=[3_+ 1 l Lp (1 - 2Lp) .

(13)

We can obtain an expression which gives the value of Lm as a function of f.:.

Lm = { (f.:(3 + 2G) - 3)

+ [(};.(3 + 2G) - 3)2

- 8Gf.:(l - f.:)]I 12 }(4Gf.:)- 1 , (14)

*'* 4P

so finally Eq. 12 is parameterized in terms of f.: instead of Lm.

This formulation can easily be generalized to mixtures of ellipsoidal (general, three-axis ellipsoid) particles hav­ing different shapes (i.e., different axial ratios) or differ­ent dielectric constants. In the most general case, the ex­pression is

(E) = [o _·f) ± Em k=l (1 - Lm){E) + Lm,Em

+ i ± ft EPkl l t=l k=l (1 - Lp,){E) + Lp,,Ep" (15)

[ 3 1

. (1 -f) 2: ------k= I (1 - Lm,)( E) + Lm, Em

+ ~ ~ ft (1 - Lp,)(~) + Lp,,EJ-1

where subscript l runs over the different shapes or di­electric constants, and where the morphological axes of the ellipsoids are taken to be coincident with the principal axes of the dielectric tensor. If we hold by the restriction to optically uniaxial spheroids, the percolation threshold is given by

H f.:= H + G' (16)

In this case G' takes the value

(17)

where ck = ftJf, and l ftc = 1 (cf. Eq. 10 and following paragraph).

Equation 15 must be solved by an iterative procedure. Some simplification can be expected for the low concen­tration case limit; at this limit the concept of percolation threshold disappears and the dielectric constant can be calculated by

( ) [ ~ ft ~ f ( EPtk - Em) ll E = E 1 + L_. - L_. (18) m /=1 3 k=l (1 - LPt,) Em + LP,EPtk .

This result can also be obtained if we expand the MGT expression generalized for ellipsoidal particles by Hay­ashi et al. 10 into a Taylor series in powers off; this ex­pression has been used on several occasions6 to calculate IR absorption spectra of diluted systems.

RESULTS AND DISCUSSION

In order to apply the above ideas, we have measured a series of absorbance spectra of different substances to test the role that aggregation and crystallinity effects play in the IR absorption spectra of powdered samples. In the following paragraphs we will to try to quantify all these effects in well-known systems such as MgA120 4, MgO, a-Fe20 3, and Ti02 • Proofs of the influence of crystallinity in the IR spectra will be given in the systems KBr/ MgA120 4 and KBr/a-Fe20 3 • Aggregation effects have been detected previously in MgO/KBr, where several au­thors (see, for instance, Ref. 4) have verified that only after exhaustive grinding and shaking of the matrix-par-

APPLIED SPECTROSCOPY 1557

/

!·

44.00

300 400

44.40 44.80

28 45.20

500 600 700 800 Wavenumber (cm- 1)

900 1000

FIG. 6. Data (arbitrary units) for three different samples of spinel hav­ing different degrees of crystallinity, resulting from annealing at the temperatures indicated: (A) X-ray diffraction peak corresponding to the 400 Bragg reflection; (B) IR absorbance spectra.

ticle mixture before pressure-sintering will the final spec­trum approach the calculated one. However, large dis­crepancies are still observed in the absolute absorbance scale.

Spinel, MgAl20 4• We have recorded the IR spectra 9f samples of spherical particles of spinel with different de­grees of crystalline order, diluted in KBr. We have chosen this substance because it is originally obtained as an

TABLE I. Crystallite size deduced by Scherrer formula for three different spinel samples.

Annealing temperature (°C) Width at half-hei!lht (26°) Crystallite size (A)

800 0.50

520

1075 0.38

1350

I !50 0.31

3000

amorphous material (as determined by XRD), and, since it does not undergo any phase transition nor do its par­ticles sinterize in a very wide temperature range, it is possible to obtain monodispersed spheres of this material with different degrees of crystallinity depending on the annealing temperature.

The samples studied were subjected to annealing treatments lasting 3 hours at 800, 1075, and 1150 °C, respectively. Their IR absorbance spectra and XRD pat­tern corresponding to the 400 Bragg peak appear in Fig. 6. It can be noticed that the bands of the IR spectra become narrower as crystallinity improves. We have es­timated the average crystallite size in each sample by Scherrer's procedure, and the results are in Table I. We believe this to be the first explicit experimental dem­onstration of the variation of the optical constant of small particles with crystallinity, proving that this factor has to be reckoned with when explaining the IR prop­erties of solids. Crystal sizes smaller than 1000 o pro­duce broader and less intense bands; above this value, the optical properties of small particles are nearly in­dependent of the crystal size. We have also observed the same behavior in other substances, some of them ex­haustively studied in IR as a-Fe20 3 and MgO. However, differences in crystallinity do not suffice to explain the large discrepancies between theoretical and experimen­tal spectra.

MgO. Magnesium oxide is probably the most studied insulator substance in IR absorbance by small parti­cles.2.4.9·27 However, in spite of the fact that all features of the IR absorbance spectra of small particles of MgO have been perfectly explained, it has been impossible to justify the lack of agreement between theoretical and ex­perimental absolute absorbance.

Normally, small, cube-shaped particles of MgO are ob­tained by burning magnesium ribbon in air. These small cubes tend to aggregate to make chain-like clusters, even inside the KBr pellets. To minimize this effect, Bohren and Huffman4•13 used intensive grinding and shaking of the mixture KBr/MgO (in some cases, the milling process took one week). The frequencies of the experimental maxima of these IR absorption spectra were fitted very well by the theoretical ones, but there remained a factor of 2 between experimental and theoretical extinction cross section. We attribute this lack of agreement to the effect described in Eq. 6 according to which it is very difficult to obtain a very low value of the effective filling factor for mixtures of two different kinds of particles with very different sizes.

In order to find the origin of the discrepancies between calculated and experimental spectra, we have made a new set of measurements of small cubic MgO particles dis­persed in KBr (Fig. 7 A). We prepared two different sam­ples with the same concentration of MgO (f = 0.007). One pellet was pressure-sintered without any treatment, while the second was ground for one hour prior to press-

/ 3.5 A

3.0

2.5

<U g 2.0 <tS -e 0

$ 1.5 <

1.0

0.5

<U u s:: <tS -e 0

"' .c <

400 500 600 700 800

Wavenumber (cm-1)

3.5 B

-3.0

2.5

2.0

1.5

1.0

0.5

0.0 300 400 500 600 -1 700 800

Wavenumber (cm-1)

FIG. 7._ IR _absor?ance_ spectra of MgO particles in KBr (theoretical filling factor !.n = 0.007) for (A) cubic-shaped MgO particles (60%) and sphermds With axial ratiO r = 3.5 (40%); (B) spherical MgO particles: (a) pressed without any previous grinding. (i3) KBr/MgO mixture ground for one hour and subsequently pressed. (-y) Quantitative calculation of the absorbance of a KBr/MgO mixture for an effective filling factor off. = 0.05. •If

ing. In the first sample (a in Fig. 7A), in which MgO particles are strongly aggregated, the absorption maxi­mum is placed close to the transversal frequency of sin­gle-crystal MgO (wT = 400 cm- 1), while in the ground sample (13 in Fig. 7 A) the maximum absorbance is at w = 550 cm- 1, the same frequency as reported by Fuchs,9

and Bohren and Huffman.4 There are also some peaks at higher frequencies, as Fuchs calculations for cubic par­ticles predict. It must be pointed out that, even in this sample, a strong absorption maximum still remains near the frequency of the transverse mode of the single crystal, indicating that some elongated clusters of cubic particles of MgO still persist.

We have tried to fit this spectrum assuming a mixture of cubic particles (60%) and prolate spheroids (40%). A filling factor of f.ff = 0.05 has been used (about 7 times

J,h). The optical constants of these MgO particles were obtained from specular reflectance measurements on pressed pellets of the particles, 19 and these were used in the calculations instead of the single-crystal constants. On the other hand, it has been necessary to rework Eq. 15 to make it appropriate for the calculation of the di­electric properties of cube-shaped particles embedded in a matrix. After Fuchs9 the depolarization field of a cube can approximately be written as a sum of six different terms, instead of the two which are necessary for the exact expression for a spheroid. For isotropic cubes, Eq. 15 becomes

TABLE II. Values of the six main depolarization coefficients for cube-shaped particles.

0.44 0.214

0.24 0.297

0.04 0.345

0.05 0.440

0.10 0.563

0.09 0.706

+ ± h ~ Cp,, ]-'. (lg) 1=1 k=l (1 - Lp,,)(e) + Lp,,ep,

This equation reduces back to Eq. 12 if MP = 2, and C1 = 2/3, C2 = 113; if inclusions have cubic shape, MP

6. The six C coefficients,9•28 after the Fuchs approxi­mation, are given in Table II.

It can be seen in Fig. 7 A that there still exist some problems in representing correctly the lowest frequency band, which is attributed to the presence of clusters of spheroidal particles. One probable explanation is that we are dealing with a continuous distribution of spheroidal shapes instead of only one type. However, the agreement in absolute absorbance is excellent. This is because we have used a theory to calculate average dielectric con­stant that accounts in a realistic way for aggregation ef­fects.

It is also possible to synthesize small (polycrystalline) particles of MgO with spherical shape.8 Following the same procedure as before, two different pellets were pre­pared with an absolute concentration off= 0.007. The first mixture, KBr/MgO, was pressed directly without any additional process (its spectrum is a in Fig. 7B), while the second one (13 in Fig. 7B) was ground for a period of 1 hour. It can be seen that these spectra are totally

APPLIED SPECTROSCOPY 1559

, ,.~_:~~e,x.lltj.f;/S.. _. ... , ;·~· --~~- . '

. -· -- .. ________________ ............. ;~ .... ~~···""'jjot:_,

2.5 2.5 r----------~

a b 2.0 2.0

<I) 1.5 <I) 1.5 u u :::: § ro ..0 -€ ... 0 0 til til

..0 ..0 < 1.0 < 1.0

0.5 0.5

J~tJV ~

0.0 ~ 0.0

7_vu~ ~ 250 450 650 850 250 450 650 850

Wavenumber (em-1) Wavenumber (em-1)

FIG. 8. IR absorbance spectra of spherical u-Fe20 3 particles dispersed in KBr (J;. = 0.0012). Thick line represents the observed spectrum of the pellet; thin line depicts calculated absorbance of a KBr/u-Fe20 3 mixture. Arrows indicate shoulders corresponding to strongly aggregated areas. (a) Pellet pressed without prior grinding. Absorbance calculated with the following parameters: r = I (axial ratio), f 1 = 0.19, c1 = 0.156; f 2 = 0.04, c2 = 0.682; and f 3 = 0.002, c3 = 0. 182. (b) Pellet pressed after grinding and shaking. Absorbance computed with the following parameters: r = 1,/1 = 0.15, c1 = O.l;f2 = 0.04, c2 = 0.15; andf3 = 0.001, c3 = 0.75.

different from those which appear in Fig. 7 A. In this case the absorption peak is just at the position corresponding to a surface mode for spherical MgO particles. However, due to the low crystallinity, even the better dispersed pel­let presents an IR absorbance which is nearly one-half of the value the theory predicts (-y in Fig. 7B). In this case, we can say that the dielectric properties of this sample with low crystallinity (average single-crystal size is about 50-100 A) are different from those of the single crystal (greater than 2000 A). It must be added that the fit was carried out with the optical parameters used for the cal­culation shown in Fig. 7 A, because we did not have the necessary amount of this sample to make a reflectance measurement and determine its own optical parameters.

a-Fe20 3• Like MgO, a-Fe20 3 is a substance very well studied in IR absorption spectroscopy.5·6•18 With the use of this anisotropic substance, it was experimentally shown for the first time that one and the same substance could present very different IR spectra depending on par­ticle shape.5·6 As with MgO, we made two pellets in which the absolute concentration of a-Fe20 3 was known. The a-Fe20 3 particles used had a spherical shape with a very high degree of crystallinity (Fig. 3). In Fig. 8 it is shown that the difference between ground and unground pellets is not so appreciable as in MgO. We have fitted both spectra using the IR optical constants deduced fro"m single-crystal measurements29 and adopting the hypoth­esis of heterogeneous aggregation (Eq. II). For the un-

ground pellet, three different filling factors have been used to fit the observed absorbance:f1 = 0.19, c1 = 0.156;

/ 2 = 0.04, c2 = 0.682; and/3 = 0.002, c3 = 0.182. Taking into account that the real filling factor is J,h = 0.0012, it appears that inside the pellet there must exist regions with a very high concentration in iron oxide. The weak inter­action between particles could explain such high values of the filling factor. It should be pointed out that the ef­fects of aggregation are clearly visible in the IR absorp­tion spectra as two shoulders at 320 and 460 cm- 1 (two arrows mark those shoulders in both parts of Fig. 8), below the corresponding Frohlich modes. We have made the same calculation to reproduce the spectrum of the ground pellet, and the effective filling factors used were:

/ 1 = 0.15, c 1 = 0.1;/2 = 0.04, c2 = 0.15; and/3 = 0.001, c3 = 0.75. In this case, about 75% of the particles seem to be well dispersed, but very disperse particles coexist with more concentrated clusters, as indicated by the fact that the experimental bands are wider and stronger than the theoretical ones. However, it should be borne in mind that a small mismatch between the IR single-crystal op­tical constants and those of the powder particles could also be the origin of this lack of agreement. As we have pointed out before for MgO, we have also observed that different degrees of crystallinity mean different optical parameters of the material. The IR absorption spectra of two samples of dilute small spheroids of this material, with an axial ratio r = 6.3 (one of them untreated and

300 400 500 600 700 800 Wavenumber (cm-1)

FIG. 9. IR absorbance spectra of spheroidal (axial ratio r = 6.3) a-Fe20 3 particles dispersed in KBr: (a) Spectrum of sample with poor crystallinity. (b) Spectrum of sample after annealing at 500 oc for 2.5 days. (c) Calculated spectrum of a-Fe20 3 , spheroids with r = 6.3 and f = 0.05. Absorbance scale is arbitrary.

the other annealed at 500° for 2.5 days), are compared in Fig. 9 with the calculated spectrum obtained by using in Eq. 15 the dielectric constants measured from a single crystal. It can be seen that in the· annealed sample the two characteristic bands of acicular iron oxide in the re­gion of 500-600 cm- 1 are sharp and well-resolved, as compared with the spectra of the untreated sample.

Ti02• Finally, we want to show by an example that MGT may be inadequate to explain the IR absorbance spectrum in some instances. We have chosen a material with a very high dielectric constant because in this case the differences between Eq. 15 and MGT to calculate the average dielectric constant are greater. We have prepared two pellets of Ti02 (rutile), which shows behavior at IR frequencies that is very well known7·30 and which has a very high dielectric constant in the reststrahlung re­gion.31·32 As we did with MgO, we have prepared two different KBr pellets with a precise knowledge of the filling factor (f = 0.003). One of them was extensively ground before pressing. In Fig. 10 it can be noticed that this sample has experienced a very strong increase in total absorbance and a narrowing of the main bands. However, no appreciable shifts in the maxima of the bands are observed, as theory predicts, for low-filling­factor samples. We have tried to fit these spectra with MGT and the generalized expression of effective medium theory (Eq. 15). In both cases an effective filling factor higher than the theoretical one has been necessary in or­der to fit the spectrum of the unground sample, but the filling factor calculated by MGT (data not shown) is greater than that of Eq. 15. The effective and the theo­retical filling factors calculated by Eq. 15 are very close

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400 500 600 700 800

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Fta. I 0. IR absorbance spectra of spherical Ti02 dispersed in KBr (f;. = 0.003): (a) Spectrum of sintered pellet without any previous grinding. (b) Spectrum of Ti02/KBr mixture, ground for one hour and then pressed. (c) Quantitative calculation of the absorbance of a KBrffi02

mixture for an effective filling factor f,ff = 6!, •• (f,ff = 0.017). (d) Quan­titative calculation with the same conditions as in c above, but using MGT.

for the sample that has suffered the grinding treatment. We think that this is so because these particles do not tend to aggregate as strongly as those of iron oxide, and its crystallinity is very good. So, in these "ideal condi­tions" it is possible to perform a realistic calculation of the IR absorbance spectrum. However, in most cases, it is difficult to find a material fulfilling these three condi­tions, i.e., perfect spheroidal-shaped small particles, very high crystallinity, and good dispersal characteristics. This problem could explain the lack of agreement until the present between theoretical and experimental spectra.

ACKNOWLEDGMENTS

We thank M. Ocafia and T. Gonzalez-Carrefio for furnishing some of the samples used in this work.

I. H. Frohlich, Theory of Dielectrics (Oxford University Press, Ox­ford, 1948).

2. L. Genzel and T. P. Martin, Phys. Stat. Sol. b 51, 91 (1972). 3. J. C. Maxwell-Garnett, Phil. Trans. R. Soc. (London) 203, 385

(1904); ibid. 205, 237 (1905). 4. C. F. Bohren and D. R. Huffman, Absorption and Scartering of

Light by Small Particles (John Wiley and Sons, New York, 1983), and references therein.

5. C. J. Serna, M. Ocafia, and J. E. Iglesias, J. Phys. C 20,473 (1987). 6. J. E. Iglesias, M. Ocafia, and C. J. Serna, Appl. Spectrosc. 44, 418

(1990). 7. M. Ocafia and C. J. Serna, Spectrochim. Acta 47A, 765 (1991), and

references therein. 8. C. Pecharroman, T. Gonzalez-Carrefio, and J. E. Iglesias, Appl.

Spectrosc. 47, 1203 (1993). 9. R. Fuchs, Phys. Rev. B 11, 1732 ( 1975).

10. S. Hayashi, N. Nakamori, and H. Kanamori, J. Phys. Soc. Japan 46, 176 (1979).

II. S. Y. Tong and A. A. Maradudin, Phys. Rev. 181, 1318 ( 1969).

APPLIED SPECTROSCOPY 1561

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12. L. Genzel and T. P. Martin, Phys. Stat. Sol. b 51, 101 (1972). 13. D. R. Huffman, "The Applicability of Bulk Optical Constants to

Small Particles", in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, Eds. (World Scientific, Singapore, 1988), Chan. 5, p. 279.

14. D. A. G. Bruggeman, Ann. Phys. Lpz. 24, 636 (1935). 15. D. Polder and J. H. Van Santen, Physica XII, 257 (1946). 16. R. Landauer, J. Appl. Phys. 23, 770 (1952). 17. R. Landauer, "Electrical Transport and Optical Properties of Inhom­

ogeneous Media", in Proceedings of the First Conference on the Electrical Transport and Optical Properties of Inhomogeneous Me­dia, J. C. Garland and D. B. Tanner, Eds., AlP Conf. Proc. No. 40 (AlP, New York, 1978), p. 2.

18. C. Pecharroman and J. E. Iglesias, Phys. Rev. B 49, 7137 (1994). 19. C. Pecharroman and J. E. Iglesias, J. Phys. Cond. Matter 6, 7125

(1994). 20. C. Pecharroman, T. Gonzalez-Carreii.o, and J. E. Iglesias, Phys.

Chern. Miner. 22, 21 (1995).

21. C. Pecharroman, T. Gonzalez-Carreii.o, and J. E. Iglesias, J. Mater. Res. 11, 127 (1996).

22. T. Gonzalez-Carreii.o, A. Mifsud, C. J. Serna, and J. M. Palacios, Mater. Chern. Phys. 27, 287 (1991).

23. M. Ozaki, S. Kratohvil, and E. Matijevic, J. Colloid Interface Sci. 102, 146 (1984).

24. M. P. Morales, T. Gonzalez-Carreii.o, and C. J. Serna, J. Mater. Res. 7, 2538 (1992).

25. E. Matijevic and P. Scheiner, J. Colloid Interface Sci. 63, 509 (1978).

26. D. S. McLachlan, J. Appl. Phys. 70, 3681 (1991). 27. J. T. Luxon, D. J. Montgomery, and R. Summit, Phys. Rev. 188,

I 345 (1969). 28. D. Langbein, J. Phys. A: Math. Gen. 9, 627 (1976). 29. S. Onari, T. Arai, and K. Kudo, Phys. Rev. B. 16, 1717 (1977). 30. J. T. Luxon and R. Summitt, J. Chern. Phys. 50, 1366 (1969). 31. F. Gervais and B. Pirou, Phys. Rev. B 10, 1642 (1974). 32. M. W. Ribarsky, in Handbook of Optical Constants of Solids, E.

D. Palik, Ed. (Academic Press, Orlando, Florida, 1985), p. 795.