VIBRATIONAL DENSITY OF STATES FOR LOW-DENSITY SILICA AEROGELS

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VIBRATIONAL DENSITY OF STATES FOR LOW-DENSITY SILICA AEROGELS

G. Reichenauer, U. Buchenau, J. Fricke

To cite this version:

G. Reichenauer, U. Buchenau, J. Fricke. VIBRATIONAL DENSITY OF STATES FOR LOW- DENSITY SILICA AEROGELS. Journal de Physique Colloques, 1989, 50 (C4), pp.C4-145-C4-150.

�10.1051/jphyscol:1989423�. �jpa-00229499�

R E W E DE PHYSIQUE

APPLIQUEE

Colloque C4, Supplbment au n04, Tome 24, avril 1989

VIBRATIONAL DENSITY OF STATES FOR LOW-DENSITY SILICA AEROGELS

G. REICHENAUER, U. BUCHENAU and J. FRICKE

Physikalisches Institut der Universitdt, Am Hubland, 0-8700 Wiirzburg, F.R.G.

Institut fiir Festktirperforschung. KFA Jiilich, Postfach 1913, 0-5170 Jiilich, R.F.G.

RBsumB : La densLB d'etat pour les aerogels de grande porosit6 (p = 120 et 250 ~g/rn~) a 6t6 btudi6e par diffusion inelastlque de neutrons dans un inte~alle de frequence cornpris entre 0,02 THz et 2 THz. A partir de la dependance en ternp6rature des intensites ln6lastiques entre 5 et 30

B

K, nous concluons que les excitations sont essentiellernent harmoniques (except4 pour T>200K, p = 250 Kg/m ). Les modes peuvent &re relies A des longueurs d'onde grandes par rapport aux distances atorniques puisque nous trowons un facteur de structure dynarnique identique une onde sonore. Les densk6s d'6tats normalis6es sont caiculBes B partir des intensites de diffusion inblastique totale ; elles sont donnees sur une 6chelle absolue.

Abstract

-

The vibrational density of states for highly porous silica aerogels ( p = 120 kg/m3 and 250 kg/m3) were studied with inelastic neutron scattering in the frequency range from 0.02 THz to 2 THz. From the temperature dependence of the inelastic In- tensities between 5 K and 300 K we conclude that the excitations observedare essentially harmonic (except for T > 200 K, _p = 250 kg/m3). The modes can be related to wavelengths large compared to atomic distances, as we find a sound-wave-like dynamic structure factor. The normalized vibrational densities of states are calculated from the total inelastic scattering intensities; they are given on an absolute scale.

Thermal properties of aerogels or more fundamentally, the dynamics of porous sollds can be conveniently derived from inelastic neutron scattering. Techniques as IR-absorption and Raman scattering, suffer from the fact that the transition probabilities for the various excitations are rather complex. Likewise, calorimetric measurements of silica aerogels at low temperatures are not at all trivial, as the thermal conductivity is smaller by one to three orders of magnitude compared to the bulk material. The specific heat of aerogels, on the other hand, is larger by about the same factor. Among other difficulties very large thermal time constants are to be expected.

2

-

REMARKS ON INELASTIC NEUTRON SCATTERING

A pulsed beam of neutrons with defined energy can be used in a time-of-flight experiment to probe the number of vibrational states within a specimen. As indicated in fig.1 the narrow energy distribution of the incident neutrons is broadened by scattering from the specimen; the scattered intensity versus time-of-flight consists of a sharp elastic line and a broad distribution of inelasticly scattered neutrons. The latter have excited or deexcited a vibrational mode within the specimen, and thus are detected after or prior to the elastic peak. In addition to the energy spectrum information on the angular distribution of the scattered intensity can be derived by positioning detectors at different angles. The intensity as a function of angle 8 or momentum transfer Q at fixed energy tranfer is proportional to the fourier transform of the corresponding spatial distribution of the atomic motion weighted with the scattering lengths. It thus is correlated to the kind of vibrational excitation observed.

However, an analysis is commonly only possible if the coherent scattering from the specimen can de properly separated from the total scattering which also includes incoherent contributions. Incoherent tattering occurs as the scattering length for neutrons varies strongly with mass number a n 2 nuclear spin of the scatterer; as the angular intensity depends on both, the spatial arrangement of the scatterer as well as on their scattering length, the statistical distribution of differently scattering isotopes within the specimen more or less destroys the information about the spatial correlations. The relative contributions of coherent or incoherent scattering can be influenced by reducing or enhancing the amount of *H (strongest incoherent scatterer) within the specimen during sample preparation. In porous materials with large inner surfaces incoherent scattering can be partially suppressed by removing adsorbed water (heating) or exchanging it for D,O; these procedures, however, can also change th'e structural and dynamic properties of the specimen.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989423

Starting from the'work of Carpenter and Pelizzari _{/ I / ,} U. Buchenau /2/ proposed a procedure to extract the density of states (further denoted as DOS) on an absolute scale which can be applied to the'total scattering intensity of the original sample. The main condition for its application is that the dynamic structure factor reproduces the peaks in S(Q) (the fourier transform of the pair correlation function). this condition is fullfilled automatically for vibrations with long wavelengths (long as compared to atomic distances).

3

-

EXPERIMENTAL

Inelastic neutron scattering was performed on the time-of-flight instrument SV5 (fig.1) and the triple axis spectrometer SV4 at the reactor FRJ-2 / Julich. We investigated two base- catalyzed silica aerogels of different densities (120 and 250 kg/m3); the samples were first baked at 500 OC for 40 hours and subsequently sealed under N2-atmosphere into an aluminum cylinder. The measurements were performed for temperatures between 5 and 300 K.

time of flight

-

neutron

guide

m ^/

time of flight

-

r--TT--1

Fig.1 Inelastic neuton scattering experiment: The chopped monochromatic neutron beam is scattered from the specimen and analyzed with respect to scattering angle and time-of-flight.

Elastic scattering as well as energy losses and gains due to excitation and deexcitaion of vibrational modes are observable.

4

-

^RESULTS

The elastic scattering intensity versus wave vector Q show

-

independent of the aerogel den- sity

-

two broad maxima at 1.6 A-' and about 3.0 A - l , respectively (fig.2). They correspoad to the elastic scattering profile as detected for vitreous silica / 3 / and Cab-0-Sil 141, too, At small Q values the scattering is enhanced due to small angle scattering contributions, caused by the huge inner surfaces of the aerogels.

To evaluate the data in the inelastic regime, we used the following equation (ref. 121):

with

St/1Je

the ratio of the momentum of the incident to the momentum of the scattered neutron, Q = (kc-ki) the absolute value of the momentum transfer divided by h/2n, M,the average mass of a scatterer, g(v) the DOS normalized to 3N, where N is the number of scatterers in the investigated volume and f, the Bose-factor :f, = exp(hv/k,T-1) for energy gain of the neutrons; for energy loss of the neutrons fe has to be replaced by f,+l. The integral corresponds to the elastic structure factor (fig.2) smoothed by a sound-wave-like motion of the scatterers around their equilibrium positions; i.e. q, stands for an average q-value related to the modes of frequency v. I, is the total elastic scattering intensity.

+

1.0

4$f2-~i o2 -

^{aerogel (}

^{g = 250}

A

-

^m&-

momentum transfer

Q (A-')

Fig.2 Elastic scattering intensity versus momentum transfer Q (SV4-data) for o 5 K, A 100 K and o 200 K; the elastic intensity is corrected for multiple scattering. Dashed line: elastic scattering from vitreos silica (see e.g. ref.181).

@iJ

aerogel ( = 250 kg/m3 T = 120K

,

I

,

0 %

,

/ __ - -

- - _---

__-'Lvitreous silica

momentum transfer a(A-'I

Fig.3 a) Inelastic scattering intensity versus Debye-Waller-factor exp(-2W) times Bose-factor (2W = 113 <u2> Q?, where <u2> is the mean square displacement of a scatterer within the speci- men) for modes of a mean frequency of v = 0.04 THz and an average Q-value of 1.4 A-* (for the 250 kg/m3 aerogel). A straight line would imply that the modes observed are harmonic.

b) Experimental and theoretical (-) scaled inelastic scattering intensity versus momentum transfer Q for the 250 kg/m3 aerogel at different frequencies: o 0.09 THz, o 0.26 THz, A 0.45 THz, A 0.70 THz, o 1.07 THz; dashed line: inelastic scattering from vitreous silica (ref.181; the two curves are not scaled relatively to each other).

Equation (1) holds if the modes investigated are harmonic and the model for the dynamic structure factor (- Q2 jIo(Q+q,) dz) is applicable. These two conditions are checked in fig.3 for the 250 kg/m3 aerogel. As equation (1) gives only the relation between the DOS and the one-phonon cross section, we have to correct the measured intensity for multi-phonon scattering; i.e. we have to calculate.the probability for the scattering processes, where simultaneously two, three or more modes are excited or deexcited. The multi-phonon terms can be easily calculated from the one-phonon DOS by successive convolution procedures. Thus, we have to solve our problem iteratively: We start with a model DOS (e.g. the W S extracted from the experimental data, neglecting multi-phonon contributions), determine the multi-phonon terms, subtract them from the experimental data and calculate from the remaining one-phonon part a new DOS etc. This is a numerical problem which can be simplified, when the DOS is approximated by a function whose convolution can be calculated analytically. As model function for the DOS we used a sum of a sharp lorentzian and a broad gaussian function times v 2 (fig.4).

One should point out, that multi-phonon terms caused by the DOS below the frequency resolution of the instrument gives integral contributions to'the intensity in the frequency range covered by the measurement; i.e. we even can extract some information on the DOS for frequencies below the resolution limit.

Following the above procedure the experimental data were evaluated for 0.02 THz < v < 2 THz and 1.1 A-l < Q < 2 A-'. The results for the two aerogels are compared to the DOS of vitreous silica in fig.5. The DOS of the model function in the low frequency limit was

-

by choice of the parameters for the lorentzian

-

forced to reproduce the Debye-DOS as given by the corre- sponding longitudinal and transversal sound velocities (v,, v,). Generally, the DOS should depart from the Debye-behaviour if the wavelength of the phonons (related to v by h = v/v, where v is the average sound velocity, with 3/v3 = (l/ve3 +2/ve3)) is comparable to the size of the inhomogeneities in the material. The frequency v,,, where DOS/v2 takes the value 0.5 DOS,,,,,/v2 characterizes the crossover v,, from phonons to localized states. Best consistency between the model function used to calculate the multi-phonon contributions and the final DOS was achieved for the v,, values given in table I.

Table I: The average sound velocities v are calculated from v, and v, values measured with ultrasonic transducers; ' R. Vacher, private communication, ** values, calculated from v,,, second row.

(a. u.

)

Fig.4 a) Frequency dependence of the analytical model function g(v)/v2, with L = extremly narrow lorentzian and G = broad gaussian function;

b) corresponding log-log-plot of the model DOS versus frequency with v,, being the cross-over frequency.

5

-

DISCUSSION

Generally, the low frequency W S of amorphous materials is determined by the Debye contribution and additional states as caused by two-level system / 5 / etc. As the Debye DOS in porous silica is increased by a factor of 10" to 10: due to its small sound velocities, additional states could not be detected even if they are present. Tpus, the W S in the low frequency range should be well described by the average sound velocity. The departure of the DOS from a Debye-behaviour is.characterized by the cross-over frequency v,,. If we compare the values extracted from our data to those of ref. / 5 / , we find that they are both larger by about a factor of two; this, however, can be explained by the different "definitions" for v,,.

For the range above the cross-over frequency theoretical predictions are available only for

s t r u c t u r e s which d i s p l a y a f r a c t a l behaviour. SANS-studies / 6 / a s well a s high r e s o l u t i o n EM images / 7 / of base-catalyzed a e r o g e l s i n d i c a t e , t h a t t h e s e m a t e r i a l s can not be denoted a s f r a c t a l s . The frequency range below 0.05 THz i s s e n s i b l e t o t h e mass d e n s i t y of t h e m a t e r i a l , whereas f o r frequencies above 0.1 THz t h e DOS obviously does not depend on t h e d e n s i t y of t h e a e r o g e l s . The modes corresponding v <.0.05 THz should t h e r e f o r e be r e l a t e d t o t h e porous net- work i n t h e 10 t o 100 nm-range, which determines t h e t o t a l pore volume and t h u s t h e macro- s c o p i c d e n s i t y of t h e g e l s . A s f o r d i f f e r e n t d e n s i t i e s t h e microscopic s t r u c t u r e i s s i m i l a r , we conclude t h a t t h e frequency range above 0.1 THz r e f l e c t s t h e modes of t h e 1 t o 10 nm- subunits. The l a t t e r modes a r e found t o be harmonic over t h e whole temperature range f o r both samples. A s the. experimental dynamic s t r u c t u r e f a c t o r i s i n good agreement with t h e t h e o r e t i c a l sound wave s t r u c t u r e f a c t o r , t h e e x c i t a t i o n s observed should correspond t o wavelengths l a r g e compared t o atomic d i s t a n c e s . On t h e c o n t r a r y , low frequency modes i n v i t r e o u s s i l i c a , which a r e r e l a t e d t o r o t a t i o n a l v i b r a t i o n s , show a very d i f f e r e n t s t r u c t u r e f a c t o r ( f i g . 3 b / 1 8 1 ) .

lo-& lo-3 lo-2

10-1 1 2 4

lo-& lo-3 ' ' 0 1

i

2

L

frequency V(THz1 frequency

v

(THz)

Fig.5 Normalized v i b r a t i o n a l d e n s i t y of s t a t e s g ( v ) f o r a e r o g e l s of two d i f f e r e n t d e n s i t i e s ; a ) g ( v ) c a l c u l a t e d f o r p = 250 kg/m3 and 120 K; b) d e n s i t y of s t a t e s f o r p = 120 kg/m3 a t 300 K. The f u l l curves i n both p l o t s g i v e t h e DOS-model f u n c t i o n ; f o r low frequencies t h e f u n c t i o n s show t h e Debye-behaviour g(v)-v2. The p l o t i n t h e r i g h t of f i g . 5 a ) r e p r e s e n t s t h e d e n s i t y of s t a t e s f o r v i t r e o u s s i l i c a (ref.131).

P o s s i b l e o n s e t s f o r f u r t h e r i n t e r p r e t a t i o n s a r e s m a l l - p a r t i c l e e x c i t a t i o n s i n c l u d i n g surface- s o f t e n e d modes a s discussed by B a l t e s and H i l f 191 and R i c h t e r and P a s s e l 1 141 i n t h e high frequency range, and l o n g i t u d i n a l a s well a s t r a n s v e r s a l modes of chains with d i f f e r e n t boundary c o n d i t i o n s i n t h e low frequency region j u s t above v,,.

V i b r a t i o n a l modes i n s i l i c a g e l s were a l s o s t u d i e d by Boukenter e t a l . / l o / using low f r e - quency Raman s c a t t e r i n g . In t h i s work d r i e d g e l s with d e n s i t i e s between 90 and 1700 kg/m3 were i n v e s t i g a t e d i n t h e frequency range from 0.3 THz t o 5.5 THz. I n a l l c a s e s a s c a l i n g of t h e Raman-reduced i n t e n s i t y w i t h frequency was found. The s c a l i n g exponent was r e l a t e d t o t h e s p e c t r a l dimension ';I by x = 3-a. The v a l u e s of x a r e 1.79 (acid-catalyzed aerogel with p

-

90 kg/m3), 1.69 (base-catalyzed a e r o g e l , p

=

480 kg/m3) and 1.39 (acid-catalyzed xerogel, p

-

1700 kg/m3); accordingly, t h e WS ( g ( v )

-.

vd-l) should s c a l e w i t h frequency a s 0.21, 0.31 and 0.61, r e s p e c t i v e l y . I f we e x t r a c t corresponding values from t h e neutron s c a t t e r i n g d a t a ( f o r 0.3 THz c v c 1 THz) we g e t a-1 = 1.4 f o r both a e r o g e l s . The remarkable d e v i a t i o n from our r e s u l t seems t o support t h e c r i t i q u e of Keyes and Ohtsuki 1111 on t h e e v a l u a t i o n of t h e Raman data.

F i n a l l y , we compare t h e s p e c i f i c h e a t C, c a l c u l a t e d from t h e EQS t o C,-values measured i n a c a l o r i m e t r i c experiment 1121. A s our experimental v a l u e s cover only t h e frequency range 0.02 THz c v < 2 THz, t h e WS f o r lower frequencies were taken from t h e model f u n c t i o n s ( f i g . 5 ) . The s p e c i f i c h e a t f o r t h e 250 kg/m3 aerogel divided by T3 is shown i n f i g . 6 . For temperatures above 0.4 K t h e s p e c i f i c h e a t IT' c a l c u l a t e d from neutron d a t a exceeds t h e values measured by Calemzuk e t a l . 1 1 2 1 by a f a c t o r of about 10; a s t h e temperature range between 0.4

and 10 K can be r e l a t e d t o t h e DOS f o r f r e q u e n c i e s from about 3 GHz t o 85 GHz, t h i s d i f f e r e n c e could p a r t i a l l y be explained by d e p a r t u r e s of o u r model from t h e r e a l DOS, e s p e c i a l l y f o r frequency below our r e s o l u t i o n l i m i t . Furthermore, i t i s not c l e a r whether t h e a e r o g e l s i n v e s t i g a t e d by Calemcuk e t a l . a r e base- o r a c i d - c a t a l y z e d g e l s and t h u s d i f f e r from o u r s i n t h e i r s t r u c t u r a l build-up. F u r t h e r measurements of i n e l a s t i c neutron s c a t t e r i n g i n t h e 2 t o 20 GHz range w i l l answer t h i s q u e s t i o n a s w e l l a s q u e s t i o n s f o r t h e k i n d s of e x c i t a t i o n s a t f r e q u e n c i e s c l o s e t o t h e c r o s s o v e r . E x c i t i n g neutron s c a t t e r i n g experiments on t h e c r o s s o v e r r e g i o n have been performed by H.Conrad; they a r e p r e s e n t e d i n t h e s e proceedings,too.

Fig.6 Comparison between t h e s p e c i f i c h e a t / T3 determined by a c a l o r i m e t r i c measurement f o r an a e r o g e l w i t h macroscopic d e n s i t y p = 270 kg/m3 1121 (--ma) and our c a l c u l a t i o n f o r t h e 250 kg/m3-sample using t h e d e n s i t y of s t a t e s (-.-); C, i n d i c a t e s t h e corresponding Debye- v a l u e s .

1 1

--.

I

-.

' " 1 I ^{c D =} I

⁴⁵

" {

^{J I kgK4}

I ' " I I

^{- -}

10 ! c,,= 18 J l k g ~ " -

'\

-

Acknowledgement We a r e g r a t e f u l t o RenC Vacher / U n i v e r s i t y M o n t p e l l i e r f o r providing i n f o r - mations on t h e c r o s s o v e r f r e q u e n c i e s of base-catalyzed a e r o g e l s . We a l s o want t o thank Michael Monkenbusch / KFA J i i l i c h f o r performing complementary measurements a t t h e SV4-spectrometer.

1

10-I

lo-*

REFERENCES

-

^\

- -

- 'a. -

=.

^{\ \}

Q ₌ 250 kg/m3 - -

: 9 =270kg/m ,A>/ ^-

**

^{. *'}

-

** '.

- _{** '.} -

: **

'... -

-

^**

^-.

- - ^{vitreous silica} -

1 g = 2200 kg I rn3

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L e t t .

2,

1205 (1987)

10'~ .

¹

1 0 - ~ 1 0 ' ' 1 2 4 10

temperature

(

K)