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A SANS investigation on absolute scale of a homologous series of base-catalysed silica aerogels
Journal of Non-Crystalline Solids 145 (1992) 128-132 North-Holland
NON-CR , ,iNiSOt,IDS
A SANS investigation on absolute scale of a homologous series of base-catalysed silica aerogels D o r t h e Posselt 1, J a n Skov P e d e r s e n a n d Kell M o r t e n s e n Department of Physics, RisO National Laboratory, DK-4000 Roskilde, Denmark
Small angle neutron scattering data for a homologous series of base-catalysed silica aerogels are presented. T h e data are brought on an absolute scale by normatisation with water. Both oxidised and untreated samples are investigated, and it is found that oxidation does not alter the overall aerogel structure, the only effect of oxidation being the removal of surface groups. It is concluded that the samples are best described as consisting of elemental building blocks with a radial size of 20 A, aggregated into clusters with a density d e p e n d e n t radius, 5-10 times larger than the building block size. The monolithic aerogel is formed by connection of these clusters. T h e existence of clusters in the aerogel structure is manifested in the data by the presence of cluster-cluster correlation effects. These effects are not observed in SANS spectra reported in the literature for neutrally reacted aerogels.
1. Introduction It is well established that variations in aerogel preparation conditions can have a profound influence on the structure of the resulting gel. Small angle scattering techniques using neutrons or Xrays have proven to be very well suited to systematically investigate these differences and to characterise the aerogel structure [1,2]. However, the possibility of bringing small angle neutron scattering (SANS) data on an absolute scale is surprisingly unexplored. Small angle neutron scattering data for three 'Airglass' aerogels [3] are presented. The samples are prepared using the same recipe (base catalysis), but with differing density: 0.145, 0.190 and 0.275 g / c m 3. Characterisation of the samples is found in table 1. The homologous gel series was also used to investigate the fractal dynamics of aerogels through measurement of the low temperature thermal properties, and a consistent analysis linking together the structural and dy-
namic properties was performed elsewhere [4]. The complete analysis is consistent with an aerogel structure showing three different regions. At small length scales, elemental particles of radial size, R, are found. The particles are aggregated into clusters with size, (, and the gel is formed by connection of the clusters, i.e., at large length scales, the gel is a homogeneous, porous glass. The clusters possess internal structure that can be described in fractal terms (intermediate scales). The picture of elemental particles forming fractal clusters is frequently used in analysis of structural aerogel data [1,2]. However, our study appears to be the first to include cluster-cluster correlation effects in the model used to analyse the SANS data. The presence of these effects means that individual clusters can be distinguished in the base-catalysed aerogel structure, by contrast with what is found for neutrally reacted gels [2].
2. Experimental procedure
t Present address: I M U F A , Roskilde Univel:sity Center, DK4000 Roskilde, Denmark.
The samples were cut from blocks and sanded down to a typical thickness of 2-3 mm. A set of
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
D. Posselt et al. / Homologous series of base-catalysed silica aerogels
129
Table 1 Characterisation of base-catalysed silica aerogels with densities as indicated p ( g / c m 3)
r
0.145 0.190 0.275
4.08 4.08 4.07
V (%)
L,L (m/s)
vT (m/s)
D (A)
30 47 75
182 252 423
107 158 268
310 + 4 260 _+3 211 + 3
~1
~2
(A)
(A)
R (A)
66 +_ 1 61 + 1 43 _+ 1
124 + 2 114 _+2 85 _+2
18.8 + 0,4 17.7 + 0.4 19.0 _+0.4
p, density; r, water concentration; V, dilution parameter; UL, u T, longitudinal and transverse sound velocity, respectively; D, hard core length; ~:i, ~2, cluster sizes; R, radial size of elemental particles. r and V(%) are parameters governing the production process, which is based on water, tetramethoxysilane (TMOS) and alcohol (ROH). r = [ H 2 0 ] / [ T M O S ] and V ( % ) = VTMOS/(VTMOS + VROH), where V is the volume added. The longitudinal sound velocity u L and the transverse c,T are measured by courtesy of J. Gross and J. Fricke, Universidit Wfirzburg. D, Sel, ~:2 and R are fitting parameters as described in the text. T h e value of df was fixed to 2.2 in these fits [13] (a 10% change of df does not lead to significant changes in R and D). D, ~2 and R are cited from a fit using S 2.
samples was oxidised in an oxygen atmosphere at 550°C, another set was measured as prepared. SANS experiments were performed at the Ris0 SANS facility, which covers the q-range 0.0020.55 ~ - 1 . A 6 m source-sample distance was used together with a sample-detector distance varying between 1 and 6 m, and wavelengths of o 3.0, 7.8 and 15.2 A, respectively. The spectra were overdetermined by using more settings of the spectrometer than needed to cover the full q-range. Spectra with 1 - 3 % counting statistics were recorded in 51 h. The samples were mounted directly in the beam, i.e., background spectra were measured with an empty beam. The detector dark current was estimated with a piece of boronated plastic at the sample position. Water spectra from 1 mm H 2 0 in a quartz cuvette served to eliminate differences in the detector efficiency and to bring the data on an absolute scale. Background and dark current counts were subtracted pixel by pixel properly normalised and taking sample transmissions into account. The absolute scattering cross-section, d o - / d ~ , was determined according to the formula [5]
hess d s. g is an empirical factor taking into account the deviation from isotropic scattering for the water sample due to inelastic effects and multiple scattering [6]. The spectra were azimuthally averaged (isotropically scattering samples) and, finally, spectra obtained using different settings of the spectrometer were scaled together using scale factors equal to one within a few percent.
10 3
1 0 2
~
o o
°oG°q)~'~ 0c,0oc0~
10-1 :0 '
,0 :
:o,
Io0
q [1-~]
Is 1 1 n(d~r/dX2) = ~wwTcuvg-~ (1 - Tt~20) d~TTTTTT,~
10 -2
(1)
where n is the number density of scatterers, I s and I w are the background corrected scattered intensity for the sample and water, respectively, TCuv is the transmission of the cuvette holding the water sample, TH20 is the water transmission and Ts is the transmission of the sample with thick-
r
1 0 -3
i
i
ll,i
[
10 -2
'
i
,
,
',,,
q [k-']
1 0 -1
P
10 °
Fig. 1. SANS spectra on absolute scale for oxidized samples. o, 0.145 g/cm3; zx, 0.190 g / c m 3 ; ~ , 0.275 g / c m 3. W e find n ( d o - / d ~ ) from eq. (1) and, dividing with the aerogel density, we obtain the scattering cross-section per gram of sample. The inset shows a typical fit to the model, eq. (2). In the example, the structure factor S 2 was used in a fit to the data for the oxidised 0.145 g / c m 3 sample.
D. Posselt et al. / Homologous series of base-catalysed silica aerogels
130
3. Results
Figure 1 shows the absolute scattering crosssection per gram of sample for the oxidised sample series. In fig. 2, scattering curves for oxidised and untreated samples are compared. Oxidation only influences the spectrum at high values of q, where scattering from individual building blocks is observed. The observed change upon oxidation is a decrease in the incoherent background with roughly a factor 2. The incoherent background is due to hydrogen bound in organic groups and hydroxyl groups to the surface of the elemental particles. These groups are removed by oxidation, consistent with the lower incoherent background for oxidised samples. Thus, we conclude that oxidation leaves the overall aerogel structure unchanged.
4. Discussion
From the behaviour of the scattering cross-section shown in fig. 1, we immediately conclude that there exists a characteristic q-value for each density, above which the scattering curves are indistinguishable, i.e., the structure of each sample has a characteristic length, ~(p), below which the sample structure is density-independent. In
102
I
I
0.145g/cm s O. 1 9 0 g / c m a 10°
0.275g/cm ~ s~
"~
.~ S
L ~~
d~r/d~Q
~.~ 1 0 - 2
1 0 -4 1 0 -5
our interpretation, this length corresponds to a cluster size. In the region of common behaviour, a 'kink' is seen on the scattering curve at q ~ 0.08 o 1 1. For q > 0 . 0 8 A , a Porod behaviour is observed (dcr/dX? = 2,rrA2Sq -4, where B is the scattering length density and S is the specific surface area [7]), while the data below 0.08 A-1 roughly follow a power-law with an exponent ~ 2.1. The Porod behaviour is consistent with scattering from individual building blocks; the exponent - 4 indicates a smooth surface of the elemental particles. From the absolute scattering cross-section, we calculate an average surface area of 370 ma/g, assuming that the density of the building blocks equals the density of amorphous silica (2.2 g/cm3). The region below 0.08 A-1 is 'traditionally' interpreted in a fractal picture. Due to the relatively large density of the investigated gels, this power-law region is rather small, i.e., ~/R is relatively small. In setting up a model for the scattering cross-section, our basis is thus the following. At high values of q, a particle form factor describes the scattering cross-section, and at small values of q the small angle pattern can be considered the result of scattering from a collection of clusters. The internal structure of the clusters dominates the scattering pattern in the intermediate fractal range. We interpret the 'bump' seen in the spectra, when crossing over into the flat part of the spectrum from the powerlaw region, as being due to cluster-cluster correlations. It is well known that such effects lower the intensity at small values of q [8]. Our model for the scattering cross-section is based on the basic expression
I
I
1 0 -2
1 0 -s
10 °
Q [/~-1] Fig. 2. Comparison between SANS spectra for oxidised and untreated samples, o, oxidized; o, untreated. For clarity the intensity scale is shifted for each density.
=Af2( q)S( q)dP(q) + B,
(2)
where A is an amplitude proportional to particle concentration together with the volume and contrast squared; f2(q) is the elemental particle form factor, the structure factor S(q) describes the correlation between particles in a cluster, and @(q) takes account of the cluster-cluster correlations. B gives the incoherent background. We assume the elemental building blocks have an average radial size, R, and in a first approximation they can be described as spheres. An analytic expression for a sphere form factor is well known,
D. Posse# et al. / Homologous series of base-catalysed silica aerogels
but gives rise to oscillations in the theoretical spectrum. These ,~scillations could be smeared by including resoluL~., and polydispersity, but we chose instead to use an expression with the correct limiting behaviour: 1 f 2 ( q ) _ 1 + k ( q R ) 4"
(3)
The constant k is chosen so the expression at qR = 1 equals the expansion of the sphere form factor giving the Guinier expression [9], i.e., k = 0.22. An expression, S~, frequently used for the structure factor describing the internal structure of fractal clusters, exists in the literature [10]. This approach however has the disadvantage of working with a definition of the characteristic length, s~, which is not easily related to a real cluster size. As an alternative, we have in addition used an expression analogous to the particle form factor: C S2 = 1 +
1 + 0.22(qs~2) de'
(4)
where df is the mass fractal dimension, giving the scaling of mass with distance. This approach has the advantage that the characteristic length can be directly compared with R. However, an additional fitting parameter, C, is introduced• In the first model, S1, the parameter corresponding to C is eliminated by normalisation [11]: 1 d f F ( d f - 1) S~(q) = 1 + - ( q R ) df (1 + ( 1 / q ~ 1 ) 2 ) (de-I)/2
×sin[(df-
1) t a n - l ( q ~ : i ) ] ,
(5)
where F is the gamma function. The effect of the sample being a highly concentrated collection of clusters is treated in a hard sphere model, following the analysis given in ref. [12], the final expression being 1 4a(q) = 1 + p O ( q D )
sin(qD) - qD cos(qD) O(qD) =3
The cluster-cluster correlations thus enter our model described by two parameters: p, which is proportional to the packing ratio of clusters ( p ~ 5 for random close packing of spherical clusters); and D, a hard core length, which is a measure of the cluster diameter. Our full model is described by seven (using S 1) or eight (using S 2) parameters: A, p, D, {:, dr, R, B and possibly C. A typical fit is shown in the inset in fig. 1. Each of the parameters is determined by a rather small part of the scattering curve. This problem is most severe for the fractal parameters df and {:, not only because ~ / R is small, but also because the cluster-cluster correlation effects distort the part of the curve that is most sensitive to the value of {:, i.e., the rounding of the structure factor is completely annihilated by the reduction in intensity due to cluster-cluster correlation effects. The hard core length, D, is, by contrast, well determined by the 'bump' in the scattering curve, and this length is expected to scale with s~. However, D cannot be expressed in terms of E1 (eq. (5)), because this length is defined through a convenient choice of exponential decay of the fractal correlation function. For the length ~a on the other hand (eq. (4)), it is possible to compare directly with D / 2 , because in this case both lengths are measures of a cluster size. The fitted values of D and R change only few percent, when the values of the fractal parameters ~: and df are changed, and it is thus these values which are the main outcome of our analysis. We cannot assign a fractal dimension with great certainty, but the combined evidence points to a value of 2.2-2.4 [13]. The use of S z gives slightly lower X 2 values than use of S I (typically X 2 is in the range 5-15; the larger the value, the higher the density of the sample). For all fits, the packing parameter, p, has a value of approximately 4, which is in reasonable agreement with the random close packing value of 5. Within 5%, we obtain the same value for R for all sample densities. The average value of R is 20 A; 10% higher, using S 1 and 10% lower, using S 2. A systematic decrease with increasing density is found for all cluster length scales; the ratio of a fitted low density length to a fitted high density length is for all fits ~ 1.5. S z was cono
'
(qD) 3
(6)
131
132
D. Posse# et al. / Homologous series of base-catalysed silica aerogels
structed in order to have an analogous definition of ~: and R to estimate the fractal range # / R . For the lowest density sample we find # 2 / R = 6.6, for the middle density sample sC2/R = 6.4 and for the highest density sample the value is 4.5. D / 2 , being a n o t h e r m e a s u r e of the cluster size, is for all samples ~ 30% higher than ~2 (typical values for D, ~:1, ~:2 and R are given in table 1). Thus, we must conclude that the fractal range is less than one d e c a d e for all samples. T h e fractal c o n c e p t does not apply to our samples in a rigourous sense, but should be r e g a r d e d as describing the crossing over b e t w e e n two regions of distinctly different structure. A t small scales, the elemental particles are found, while the aerogel on large scales is a porous glass with inhomogeneities on length scales c o r r e s p o n d i n g to a cluster size. T h e crossover region does give rise to specific dynamic properties, which can be explained with reference to fractal models [4]. O u r S A N S spectra clearly d e m o n s t r a t e the close relationship between samples of different density, and one can regard the fractal description of the structure to be fully valid in the (unphysical) limit of zero density.
5. Conclusions F r o m our S A N S work we conclude that the base-catalysed aerogel samples are m a d e up of particles with a radial size of ~ 20 ,~. T h e particles stick t o g e t h e r and f o r m clusters with a radius 5 - 1 0 times larger than the particle radius, the size increasing with decreasing density. W e find that the base-catalysed aerogels are m a d e by connection of clusters and that individual clusters a p p e a r in the aerogel structure as evidenced by c l u s t e r - c l u s t e r correlation effects. This behaviour is by contrast with w h a t is f o u n d for neutrally reacted and acid-catalysed samples, where no ' b u m p ' is observed in the S A N S spectra [2]. T h e
same study also shows the spectrum of a basecatalysed 0.195 g / c m 3 sample, which does show a bump. O t h e r studies of base-catalysed samples of very low density (0.088 g / c m 3) show no ' b u m p ' in the spectra [1], while a study of base-catalysed samples covering densities f r o m 0.02 to 0.2 g / c m 3 develop signs of what we term c l u s t e r - c l u s t e r correlation effects with increasing density [14]. It thus appears that the growth scheme for basecatalysed aerogels with a density larger than ~ 0.1 g / c m 3 constitutes a special class.
References [1] D.W. Schaefer and K.D. Keefer, Phys. Rev. Lett. 56 (1986) 2199. [2] R. Vacher, T. Woignier, J. Pelous and E. Courtens, Phys. Rev. B37 (1988) 6500. [3] Airglass, A.B., Box 150, S-245 00 Staffanstorp, Sweden. [4] T. Sleator, A. Bernasconi, D. Posselt, J.K. Kjems and H.R. Ott, Phys. Rev. Lett. 66 (1991) 1070; D. Posselt, J.K. Kjems, A. Bernasconi, T. Sleator and H.R. Ott, Europhys. Lett. 16 (1991) 59; D. Posselt, thesis, Ris0-M2975 Report, Rise National Laboratory (1991); A. Bernasconi, T. Sleator, D. Posselt, J.K. Kjems and H.R. Ott, these Proceedings, p. 202. [5] G.D. Wignall and F.S. Bates, J. Appl. Crystallogr. 20 (1987) 28. [6] B. Jacrot, Rep. Prog. Phys. 39 (1976) 911; Unpublished, Oberthiir (1982). [7] G. Porod, Kolloid-Z. 124 (1951) 83. [8] O. Glatter, J. Appl. Crystallogr. 14 (1981) 101. [9] A. Guinier, Ann. Phys. (Paris) 12 (1939) 161. [10] T. Freltoft, J.K. Kjems and S.K. Sinha, Phys. Rev. B33 (1986) 269. [11] J. Teixeira, in: On Growth and Form, ed. H. Stanley and N. Ostrowsky, NATO A8I E-100 (Nijhoff, Dordrecht, 1988) p. 145. [12] G. Fournet, Acta Crystall. 4 (1951) 293. [13] Brillouin light scattering experiments on our gels indicate df ~ 2.2 (these experiments were performed by courtesy of R. Vacher and E. Courtens on the same setup, using the same analysis as reported in E. Courtens, R. Vacher, J. Pelous and T. Woignier, Europhys. Lett. 6 (1988) 245). [14] T. Woignier, J. Phallippou, R. Vacher, J. Pelous and E. Courtens, J. Non-Cryst. Solids 121 (1990) 198.
NON-CR , ,iNiSOt,IDS
A SANS investigation on absolute scale of a homologous series of base-catalysed silica aerogels D o r t h e Posselt 1, J a n Skov P e d e r s e n a n d Kell M o r t e n s e n Department of Physics, RisO National Laboratory, DK-4000 Roskilde, Denmark
Small angle neutron scattering data for a homologous series of base-catalysed silica aerogels are presented. T h e data are brought on an absolute scale by normatisation with water. Both oxidised and untreated samples are investigated, and it is found that oxidation does not alter the overall aerogel structure, the only effect of oxidation being the removal of surface groups. It is concluded that the samples are best described as consisting of elemental building blocks with a radial size of 20 A, aggregated into clusters with a density d e p e n d e n t radius, 5-10 times larger than the building block size. The monolithic aerogel is formed by connection of these clusters. T h e existence of clusters in the aerogel structure is manifested in the data by the presence of cluster-cluster correlation effects. These effects are not observed in SANS spectra reported in the literature for neutrally reacted aerogels.
1. Introduction It is well established that variations in aerogel preparation conditions can have a profound influence on the structure of the resulting gel. Small angle scattering techniques using neutrons or Xrays have proven to be very well suited to systematically investigate these differences and to characterise the aerogel structure [1,2]. However, the possibility of bringing small angle neutron scattering (SANS) data on an absolute scale is surprisingly unexplored. Small angle neutron scattering data for three 'Airglass' aerogels [3] are presented. The samples are prepared using the same recipe (base catalysis), but with differing density: 0.145, 0.190 and 0.275 g / c m 3. Characterisation of the samples is found in table 1. The homologous gel series was also used to investigate the fractal dynamics of aerogels through measurement of the low temperature thermal properties, and a consistent analysis linking together the structural and dy-
namic properties was performed elsewhere [4]. The complete analysis is consistent with an aerogel structure showing three different regions. At small length scales, elemental particles of radial size, R, are found. The particles are aggregated into clusters with size, (, and the gel is formed by connection of the clusters, i.e., at large length scales, the gel is a homogeneous, porous glass. The clusters possess internal structure that can be described in fractal terms (intermediate scales). The picture of elemental particles forming fractal clusters is frequently used in analysis of structural aerogel data [1,2]. However, our study appears to be the first to include cluster-cluster correlation effects in the model used to analyse the SANS data. The presence of these effects means that individual clusters can be distinguished in the base-catalysed aerogel structure, by contrast with what is found for neutrally reacted gels [2].
2. Experimental procedure
t Present address: I M U F A , Roskilde Univel:sity Center, DK4000 Roskilde, Denmark.
The samples were cut from blocks and sanded down to a typical thickness of 2-3 mm. A set of
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
D. Posselt et al. / Homologous series of base-catalysed silica aerogels
129
Table 1 Characterisation of base-catalysed silica aerogels with densities as indicated p ( g / c m 3)
r
0.145 0.190 0.275
4.08 4.08 4.07
V (%)
L,L (m/s)
vT (m/s)
D (A)
30 47 75
182 252 423
107 158 268
310 + 4 260 _+3 211 + 3
~1
~2
(A)
(A)
R (A)
66 +_ 1 61 + 1 43 _+ 1
124 + 2 114 _+2 85 _+2
18.8 + 0,4 17.7 + 0.4 19.0 _+0.4
p, density; r, water concentration; V, dilution parameter; UL, u T, longitudinal and transverse sound velocity, respectively; D, hard core length; ~:i, ~2, cluster sizes; R, radial size of elemental particles. r and V(%) are parameters governing the production process, which is based on water, tetramethoxysilane (TMOS) and alcohol (ROH). r = [ H 2 0 ] / [ T M O S ] and V ( % ) = VTMOS/(VTMOS + VROH), where V is the volume added. The longitudinal sound velocity u L and the transverse c,T are measured by courtesy of J. Gross and J. Fricke, Universidit Wfirzburg. D, Sel, ~:2 and R are fitting parameters as described in the text. T h e value of df was fixed to 2.2 in these fits [13] (a 10% change of df does not lead to significant changes in R and D). D, ~2 and R are cited from a fit using S 2.
samples was oxidised in an oxygen atmosphere at 550°C, another set was measured as prepared. SANS experiments were performed at the Ris0 SANS facility, which covers the q-range 0.0020.55 ~ - 1 . A 6 m source-sample distance was used together with a sample-detector distance varying between 1 and 6 m, and wavelengths of o 3.0, 7.8 and 15.2 A, respectively. The spectra were overdetermined by using more settings of the spectrometer than needed to cover the full q-range. Spectra with 1 - 3 % counting statistics were recorded in 51 h. The samples were mounted directly in the beam, i.e., background spectra were measured with an empty beam. The detector dark current was estimated with a piece of boronated plastic at the sample position. Water spectra from 1 mm H 2 0 in a quartz cuvette served to eliminate differences in the detector efficiency and to bring the data on an absolute scale. Background and dark current counts were subtracted pixel by pixel properly normalised and taking sample transmissions into account. The absolute scattering cross-section, d o - / d ~ , was determined according to the formula [5]
hess d s. g is an empirical factor taking into account the deviation from isotropic scattering for the water sample due to inelastic effects and multiple scattering [6]. The spectra were azimuthally averaged (isotropically scattering samples) and, finally, spectra obtained using different settings of the spectrometer were scaled together using scale factors equal to one within a few percent.
10 3
1 0 2
~
o o
°oG°q)~'~ 0c,0oc0~
10-1 :0 '
,0 :
:o,
Io0
q [1-~]
Is 1 1 n(d~r/dX2) = ~wwTcuvg-~ (1 - Tt~20) d~TTTTTT,~
10 -2
(1)
where n is the number density of scatterers, I s and I w are the background corrected scattered intensity for the sample and water, respectively, TCuv is the transmission of the cuvette holding the water sample, TH20 is the water transmission and Ts is the transmission of the sample with thick-
r
1 0 -3
i
i
ll,i
[
10 -2
'
i
,
,
',,,
q [k-']
1 0 -1
P
10 °
Fig. 1. SANS spectra on absolute scale for oxidized samples. o, 0.145 g/cm3; zx, 0.190 g / c m 3 ; ~ , 0.275 g / c m 3. W e find n ( d o - / d ~ ) from eq. (1) and, dividing with the aerogel density, we obtain the scattering cross-section per gram of sample. The inset shows a typical fit to the model, eq. (2). In the example, the structure factor S 2 was used in a fit to the data for the oxidised 0.145 g / c m 3 sample.
D. Posselt et al. / Homologous series of base-catalysed silica aerogels
130
3. Results
Figure 1 shows the absolute scattering crosssection per gram of sample for the oxidised sample series. In fig. 2, scattering curves for oxidised and untreated samples are compared. Oxidation only influences the spectrum at high values of q, where scattering from individual building blocks is observed. The observed change upon oxidation is a decrease in the incoherent background with roughly a factor 2. The incoherent background is due to hydrogen bound in organic groups and hydroxyl groups to the surface of the elemental particles. These groups are removed by oxidation, consistent with the lower incoherent background for oxidised samples. Thus, we conclude that oxidation leaves the overall aerogel structure unchanged.
4. Discussion
From the behaviour of the scattering cross-section shown in fig. 1, we immediately conclude that there exists a characteristic q-value for each density, above which the scattering curves are indistinguishable, i.e., the structure of each sample has a characteristic length, ~(p), below which the sample structure is density-independent. In
102
I
I
0.145g/cm s O. 1 9 0 g / c m a 10°
0.275g/cm ~ s~
"~
.~ S
L ~~
d~r/d~Q
~.~ 1 0 - 2
1 0 -4 1 0 -5
our interpretation, this length corresponds to a cluster size. In the region of common behaviour, a 'kink' is seen on the scattering curve at q ~ 0.08 o 1 1. For q > 0 . 0 8 A , a Porod behaviour is observed (dcr/dX? = 2,rrA2Sq -4, where B is the scattering length density and S is the specific surface area [7]), while the data below 0.08 A-1 roughly follow a power-law with an exponent ~ 2.1. The Porod behaviour is consistent with scattering from individual building blocks; the exponent - 4 indicates a smooth surface of the elemental particles. From the absolute scattering cross-section, we calculate an average surface area of 370 ma/g, assuming that the density of the building blocks equals the density of amorphous silica (2.2 g/cm3). The region below 0.08 A-1 is 'traditionally' interpreted in a fractal picture. Due to the relatively large density of the investigated gels, this power-law region is rather small, i.e., ~/R is relatively small. In setting up a model for the scattering cross-section, our basis is thus the following. At high values of q, a particle form factor describes the scattering cross-section, and at small values of q the small angle pattern can be considered the result of scattering from a collection of clusters. The internal structure of the clusters dominates the scattering pattern in the intermediate fractal range. We interpret the 'bump' seen in the spectra, when crossing over into the flat part of the spectrum from the powerlaw region, as being due to cluster-cluster correlations. It is well known that such effects lower the intensity at small values of q [8]. Our model for the scattering cross-section is based on the basic expression
I
I
1 0 -2
1 0 -s
10 °
Q [/~-1] Fig. 2. Comparison between SANS spectra for oxidised and untreated samples, o, oxidized; o, untreated. For clarity the intensity scale is shifted for each density.
=Af2( q)S( q)dP(q) + B,
(2)
where A is an amplitude proportional to particle concentration together with the volume and contrast squared; f2(q) is the elemental particle form factor, the structure factor S(q) describes the correlation between particles in a cluster, and @(q) takes account of the cluster-cluster correlations. B gives the incoherent background. We assume the elemental building blocks have an average radial size, R, and in a first approximation they can be described as spheres. An analytic expression for a sphere form factor is well known,
D. Posse# et al. / Homologous series of base-catalysed silica aerogels
but gives rise to oscillations in the theoretical spectrum. These ,~scillations could be smeared by including resoluL~., and polydispersity, but we chose instead to use an expression with the correct limiting behaviour: 1 f 2 ( q ) _ 1 + k ( q R ) 4"
(3)
The constant k is chosen so the expression at qR = 1 equals the expansion of the sphere form factor giving the Guinier expression [9], i.e., k = 0.22. An expression, S~, frequently used for the structure factor describing the internal structure of fractal clusters, exists in the literature [10]. This approach however has the disadvantage of working with a definition of the characteristic length, s~, which is not easily related to a real cluster size. As an alternative, we have in addition used an expression analogous to the particle form factor: C S2 = 1 +
1 + 0.22(qs~2) de'
(4)
where df is the mass fractal dimension, giving the scaling of mass with distance. This approach has the advantage that the characteristic length can be directly compared with R. However, an additional fitting parameter, C, is introduced• In the first model, S1, the parameter corresponding to C is eliminated by normalisation [11]: 1 d f F ( d f - 1) S~(q) = 1 + - ( q R ) df (1 + ( 1 / q ~ 1 ) 2 ) (de-I)/2
×sin[(df-
1) t a n - l ( q ~ : i ) ] ,
(5)
where F is the gamma function. The effect of the sample being a highly concentrated collection of clusters is treated in a hard sphere model, following the analysis given in ref. [12], the final expression being 1 4a(q) = 1 + p O ( q D )
sin(qD) - qD cos(qD) O(qD) =3
The cluster-cluster correlations thus enter our model described by two parameters: p, which is proportional to the packing ratio of clusters ( p ~ 5 for random close packing of spherical clusters); and D, a hard core length, which is a measure of the cluster diameter. Our full model is described by seven (using S 1) or eight (using S 2) parameters: A, p, D, {:, dr, R, B and possibly C. A typical fit is shown in the inset in fig. 1. Each of the parameters is determined by a rather small part of the scattering curve. This problem is most severe for the fractal parameters df and {:, not only because ~ / R is small, but also because the cluster-cluster correlation effects distort the part of the curve that is most sensitive to the value of {:, i.e., the rounding of the structure factor is completely annihilated by the reduction in intensity due to cluster-cluster correlation effects. The hard core length, D, is, by contrast, well determined by the 'bump' in the scattering curve, and this length is expected to scale with s~. However, D cannot be expressed in terms of E1 (eq. (5)), because this length is defined through a convenient choice of exponential decay of the fractal correlation function. For the length ~a on the other hand (eq. (4)), it is possible to compare directly with D / 2 , because in this case both lengths are measures of a cluster size. The fitted values of D and R change only few percent, when the values of the fractal parameters ~: and df are changed, and it is thus these values which are the main outcome of our analysis. We cannot assign a fractal dimension with great certainty, but the combined evidence points to a value of 2.2-2.4 [13]. The use of S z gives slightly lower X 2 values than use of S I (typically X 2 is in the range 5-15; the larger the value, the higher the density of the sample). For all fits, the packing parameter, p, has a value of approximately 4, which is in reasonable agreement with the random close packing value of 5. Within 5%, we obtain the same value for R for all sample densities. The average value of R is 20 A; 10% higher, using S 1 and 10% lower, using S 2. A systematic decrease with increasing density is found for all cluster length scales; the ratio of a fitted low density length to a fitted high density length is for all fits ~ 1.5. S z was cono
'
(qD) 3
(6)
131
132
D. Posse# et al. / Homologous series of base-catalysed silica aerogels
structed in order to have an analogous definition of ~: and R to estimate the fractal range # / R . For the lowest density sample we find # 2 / R = 6.6, for the middle density sample sC2/R = 6.4 and for the highest density sample the value is 4.5. D / 2 , being a n o t h e r m e a s u r e of the cluster size, is for all samples ~ 30% higher than ~2 (typical values for D, ~:1, ~:2 and R are given in table 1). Thus, we must conclude that the fractal range is less than one d e c a d e for all samples. T h e fractal c o n c e p t does not apply to our samples in a rigourous sense, but should be r e g a r d e d as describing the crossing over b e t w e e n two regions of distinctly different structure. A t small scales, the elemental particles are found, while the aerogel on large scales is a porous glass with inhomogeneities on length scales c o r r e s p o n d i n g to a cluster size. T h e crossover region does give rise to specific dynamic properties, which can be explained with reference to fractal models [4]. O u r S A N S spectra clearly d e m o n s t r a t e the close relationship between samples of different density, and one can regard the fractal description of the structure to be fully valid in the (unphysical) limit of zero density.
5. Conclusions F r o m our S A N S work we conclude that the base-catalysed aerogel samples are m a d e up of particles with a radial size of ~ 20 ,~. T h e particles stick t o g e t h e r and f o r m clusters with a radius 5 - 1 0 times larger than the particle radius, the size increasing with decreasing density. W e find that the base-catalysed aerogels are m a d e by connection of clusters and that individual clusters a p p e a r in the aerogel structure as evidenced by c l u s t e r - c l u s t e r correlation effects. This behaviour is by contrast with w h a t is f o u n d for neutrally reacted and acid-catalysed samples, where no ' b u m p ' is observed in the S A N S spectra [2]. T h e
same study also shows the spectrum of a basecatalysed 0.195 g / c m 3 sample, which does show a bump. O t h e r studies of base-catalysed samples of very low density (0.088 g / c m 3) show no ' b u m p ' in the spectra [1], while a study of base-catalysed samples covering densities f r o m 0.02 to 0.2 g / c m 3 develop signs of what we term c l u s t e r - c l u s t e r correlation effects with increasing density [14]. It thus appears that the growth scheme for basecatalysed aerogels with a density larger than ~ 0.1 g / c m 3 constitutes a special class.
References [1] D.W. Schaefer and K.D. Keefer, Phys. Rev. Lett. 56 (1986) 2199. [2] R. Vacher, T. Woignier, J. Pelous and E. Courtens, Phys. Rev. B37 (1988) 6500. [3] Airglass, A.B., Box 150, S-245 00 Staffanstorp, Sweden. [4] T. Sleator, A. Bernasconi, D. Posselt, J.K. Kjems and H.R. Ott, Phys. Rev. Lett. 66 (1991) 1070; D. Posselt, J.K. Kjems, A. Bernasconi, T. Sleator and H.R. Ott, Europhys. Lett. 16 (1991) 59; D. Posselt, thesis, Ris0-M2975 Report, Rise National Laboratory (1991); A. Bernasconi, T. Sleator, D. Posselt, J.K. Kjems and H.R. Ott, these Proceedings, p. 202. [5] G.D. Wignall and F.S. Bates, J. Appl. Crystallogr. 20 (1987) 28. [6] B. Jacrot, Rep. Prog. Phys. 39 (1976) 911; Unpublished, Oberthiir (1982). [7] G. Porod, Kolloid-Z. 124 (1951) 83. [8] O. Glatter, J. Appl. Crystallogr. 14 (1981) 101. [9] A. Guinier, Ann. Phys. (Paris) 12 (1939) 161. [10] T. Freltoft, J.K. Kjems and S.K. Sinha, Phys. Rev. B33 (1986) 269. [11] J. Teixeira, in: On Growth and Form, ed. H. Stanley and N. Ostrowsky, NATO A8I E-100 (Nijhoff, Dordrecht, 1988) p. 145. [12] G. Fournet, Acta Crystall. 4 (1951) 293. [13] Brillouin light scattering experiments on our gels indicate df ~ 2.2 (these experiments were performed by courtesy of R. Vacher and E. Courtens on the same setup, using the same analysis as reported in E. Courtens, R. Vacher, J. Pelous and T. Woignier, Europhys. Lett. 6 (1988) 245). [14] T. Woignier, J. Phallippou, R. Vacher, J. Pelous and E. Courtens, J. Non-Cryst. Solids 121 (1990) 198.