SAXS investigation of porous nanostructures

Journal of Non-Crystalline Solids 354 (2008) 5466–5474

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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol

SAXS investigation of porous nanostructures Katalin Sinkó a,*, Viktoria Torma b, Attila Kovács c a b c

Institute of Chemistry, L. Eötvös University, H-1117, Budapest, Hungary Department of Physical Chemistry, Budapest University of Technology and Economics, Budapest, H-1521, Hungary Department of Anatomy, Cell and Developmental Biology, L. Eötvös University, H-1117, Hungary

a r t i c l e

i n f o

Article history: Received 11 December 2007 Received in revised form 12 August 2008 Available online 24 October 2008 PACS: 81.07.b 61.05.cf 61.43.Hv Keywords: Synchrotron radiation TEM/STEM Nano-clusters Aluminosilicates Aerogels Medium-range order

a b s t r a c t Fractal and aggregate structures of aerogels were investigated by small angle X-ray scattering in order to analyze the various evaluation methods of the SAXS data for porous nanostructures. Scattering data (SAXS, USAXS) for aerogels measured with laboratory equipment as well as synchrotron technique were interpreted in the terms of Guinier, Porod, Freltoft, Teixeira, and Emmerling theories. We modified the Freltoft fit in order to get information about the structure of elementary units. The performances of the evaluation programs were studied for different aerogels structures such as fractal of wide range, fractal of limited size, and aggregate systems. The evaluation of the scattering measurements resulted in fractal dimensions, sizes of the elementary units, sizes of the fractal domains or aggregates. Quality of the fits to SAXS data was characterized by a mathematical parameter and proved by TEM photography. TEM images confirmed the sizes of the elementary building units and clusters. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The sol–gel process is one of the most practical and low energy consuming preparation methods for the synthesis of amorphous, porous materials. To preserve the structural features of the gel network and the inherent porosity of the wet gel samples, they have to be dried under supercritical conditions. Hereby, shrinkage of the monolithic body during drying is minimized and porous socalled aerogels are obtained. The analysis on porous structures of the aerogels is very important in respect of their application in the field of absorption, separation, and catalysis. In principle, the structural features of aerogels in the mesoscopic regime between 1 and 100 nm can be characterized with a variety of techniques. The mostly used one is the electron microscopy; transmissionand scanning-electron microscopy (TEM, SEM) providing direct images of the samples. The structure of ill-ordered materials, such as amorphous glasses, fractal systems, or aggregate structures is difficult to describe in direct pictures, these structures can well be characterized by scattering experiments. Small-angle X-ray scattering (SAXS) is non-destructive and gives the same informa-

* Corresponding author. Tel.: +36 1 2090555. E-mail address: [email protected] (K. Sinkó). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.08.021

tion with high statistical accuracy due to the averaging over a microscopic sample volume. The aim of the present work is to analyze the applicability and the limits of the scattering techniques applied for amorphous aerogels focusing on the nanometer sized features. The focus will be to compare the various evaluation models developed for the fractal and aggregate structures. Small-angle and ultra-small angle X-ray scattering (SAXS and USAXS), and transmission electron microscopy (TEM) have been used to determine the nanostructure of aerogels. Scattering data were obtained using several instruments such as laboratory equipment and synchrotron radiation sources and were interpreted in terms of Guinier [1], Freltoft et al. [2], Teixeria [3], and Emmerling et al. [4] theories and using the simple power-law expressions in Porod‘s region. The fractal systems of aluminosilicate and aggregate structures of silicate aerogel were studied in the experiments. 2. Experimental section 2.1. Preparation methods Aluminosilicate gels (samples 1 and 2) were prepared in a one step procedure from tetraethoxysilane (TEOS) and an inorganic Al salt (Al(NO3)3  9H2O) in organic medium at 80 °C without

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catalyst. The chemical composition was 1 mol TEOS, 1 mol Al(NO3)3  9H2O, and 20 mol 1-propanol. The gelation was performed by refluxing at 80 °C for 10–11 h [5]. Silica gels (sample 3) were synthesized from TEOS in 1-propanol in the presence of a base catalyst (NaOH). The composition of the initial solutions was 1 mol TEOS, 20 mol 1-propanol, and 1 mol NaOH. This mixture was heated at 80 °C for 3 h to get an optically clear wet silica gel. The wet gels were dried under supercritical conditions. After washing with methanol, the gels were put into a steel autoclave and washed with liquid carbon dioxide for 2–3 days at 284 K, 6.0–7.0 MPa. After completion of the solvent exchange process, the temperature in the autoclave was slowly raised to 313 K, while the pressure was kept at 10 MPa. The supercritical carbon dioxide was slowly released from the autoclave by depressurizing to an ambient pressure during 20 h. After cooling to room temperature, opaque aerogel samples could be obtained.

The structure factor of the particle centers, S(q), is related to the particle pair correlation function, G(r), by Fourier transform:

SðqÞ ¼

Z

1

GðrÞ

0

sinðqrÞ 4pr 2 dr: qr

ð4Þ

For fractal-like aggregates of primary particles, Freltoft proposed the following expression for G(r):

GðrÞ ¼ dðrÞ þ F  rðD3Þ  expðr=nÞ;

ð5Þ

where d(r) defines the Dirac delta function, F is a parameter describing the nearest neighbor correlations inside the aggregates, r means the radius of particles, D indicates the fractal dimension and n is the effective cut-off length describing the decay of the fractal-like correlations (due to the finite size of aggregates or their overlapping) [2]. The structure factor obtained by Fourier transform reads

SðqÞ ¼ 1 þ 2.2. Characterization methods

C  nD 1 CðD  1Þ  sinððD  1Þ qn ð1 þ q2 n2 ÞðD1Þ=2

 arctanðqnÞÞ; Small-angle X-ray scattering (SAXS) measurements were conducted on several instruments. The laboratory equipment was operated with a 12 kW rotating anode X-ray generator and a pinhole X-ray camera with variable distance (20.5–98.5 cm) from the sample to the two-dimensional detector (Bruker, AXS, Karlsruhe). The gels were covered in vacuum tight foil. The twodimensional spectra were corrected for parasitic pinhole scattering, as well as for the foil scattering. X-ray scattering experiments were also recorded on the JUSIFA beamline of HASYLAB at DESY in Hamburg and on the BM2 beamline in the European synchrotron radiation facility in Grenoble. All the measured intensities have been normalized to a constant value of incident X-ray flux. USAXS measurements were performed on the BW4 beamline of HASYLAB at DESY, in Hamburg. The transmission electron microscopy (TEM) investigations were performed on a JEOL 100 CX TEM with a tungsten filament operating at 100 kV in the bright field mode. 2.3. Evaluation of SAXS data Different evaluation methods were used for the interpretation of the scattering data. The radii of gyration are determined from the Guinier-plot [1]. According to the calculations of Guinier, for aggregate systems consisting of small polydisperse particles, the scattering curve can be computed by the following formula:

IðqÞ ¼

A ð1 þ q2 r2 Þ2

ð1Þ

;

where q indicates the scattering vector, r is a measure of the particle radius, and I(q) denotes the scattering intensity. If the aggregated particles form infinite fractal structures the formula might be written as

IðqÞ ¼ SðqÞ  PðqÞ ¼

A B  ; qa ð1 þ q2 r2 Þ2

ð2Þ

where S(q) defines the structure factor of the arrangement of the particles, P(q) is the single particle form factor, and a denotes the dimensionality of the fractally arranged system. In the case of porous systems, the radius of gyration (Rg) may denote the radius (r) of a particle, a pore, an aggregation, or a fractal domain. 2

r ¼

r 2g 5=3:

ð3Þ

Freltoft and independently Teixeira has developed an expression for relating the scattered intensity to the fractal structure of the aggregates [2,3]. It was assumed that the silica gels are built up from small, compact particles, which form finite fractal-like aggregates.

ð6Þ

where C is a constant. As can be calculated from the limiting behavior of S(q) at q = 0, B + 1 represents an average number of primary particles per aggregate, where B ¼ CðD  1ÞCðD  1ÞnD . The parameter n is connected with the gyration of the secondary particles [2,6]:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðD þ 1Þ rG ¼  n: 2

ð7Þ

With the same assumptions, Teixeira also developed a formula that can also be applied for the evaluation of scattering data of fractal objects [3].

SðqÞ ¼ 1 þ

D

1

ðqrÞD ð1 þ 1=q2 n2 ÞðD1Þ=2

CðD  1Þ  sinððD  1Þ

 tan1 ðqnÞÞ;

ð8Þ

where D indicates the fractal dimension, r is the radius of particles, n is a ‘‘cut-off distance”, to describe the behavior of the pair correlation function at large distances [3]. Emmerling has upgraded and applied the expression of Teixeira for the evaluation of SAXS data for aerogels [4].

IðqÞ ¼ V o /ðqÞSðqÞPðqÞ;

ð9Þ

where Vo denotes the volume of primary particles, P(q) is the Debye function in this case, and S(q) is derived from Teixeira assumption. The new component, /(q) refers to the ‘‘concentration effect”, i.e. close packing of nearly monodisperse clusters [4]. There are four parameters in the Freltoft formula, and only three in the Teixeira. On the other hand, if n is large compared to R, and the reciprocal of the measured smallest q limit, the denominator of the Freltoft fit becomes large, as well as it is equal to (qR)D in the Teixeira model. This fact enlarges the difficulties of the calculation of n by Teixeira. The Emmerling expression can be reduced, if the upper-limit length-scale for the mass fractal domain (n) is taken to be infinite. An infinite size of a fractal object can be taken into account if the size of fractal domain is out of the scattering q-range. The simplified Emmerling–Fratzl formula [7]

SðqÞ ¼ 1 þ

DCðD  1Þ sin½pðD  1Þ=2 ðqRÞD

:

ð10Þ

Pedersen and Mortensen have written an expression for the estimation of the cluster–cluster correlation length [8]. The fractal dimensions can be obtained from the slope (l) of SAXS curves in the Porod’s region using a simple power-law expression; I(q)  qD [9,10]. Schaefer et al. dealt intensively with the evaluation of SAXS

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data for silica [11–15] and organic [16] aerogels. The width and the thickness of chains making up the silica skeleton of aerogels have also been evaluated from SAXS data [17,18]. In this work, we modified the Freltoft equation by introducing a new parameter: a to the particle form factor.

The original Freltoft form factor : PðqÞ ¼

and the modified Freltoft factor : PðqÞ ¼

1

ð1 þ

pffiffi 2 2 2 2 q rF Þ 3

ð1 þ

pffiffi 2 2 2 2a q rF Þ 3

1

;

;:

ð11Þ

where rF is the Freltoft radius, which the radius of elementary units p was calculated from r = rF2/ 0.22. A characteristic parameter for the fractal behavior of the elementary units can be calculated from the value of a.

du ¼ 2ð2  aÞ;

ð12Þ

where du characterizes the structure, the structural density of elementary units. The values between 2 and 3 indicate loose, fractallike or randomly branched structures for primary particles. The du = 4 denotes a compact elementary unit with smooth surface. If the value of du is between 3 and 4, it describes the elementary units as 3D particles with fractally rough surfaces. The SAXS curves for simulated data were intended to interpret the meaning of a parameter (Fig. 1). By varying the value of a, the scattering intensity of elementary units changes (in the range of 1–10 nm). Serving as an illustration, the slope of SAXS curves simulated with a = 0 (calculated du = 4) proved to be 4.06 in the assigned range. Since the various algorithms for fitting the applied functions to the measured data may land in local minima of the error hypersurface, only parameter estimates exhibiting real physical meaning were accepted. We have measured the backgrounds and the data were corrected by the absorption of samples. These values were subjected to subtract. The background proved to be independent of q in the measuring range, the relation was negligible even in the small qrange. In this manner the background can be considered as standard. When calculating the exact mathematical formula of the fitting function, the measuring limit of the instrument, as well as the constant background scattering has to be taken into account. Therefore all the curves were fitted with the following expression:

logðImeasured ðqÞÞ ¼ logðB þ Imodell ðqÞÞ;

ð13Þ

α

Intensity (a.u.)

1.5 1.2 0.9 0.7 0.5 0.2 0.0

0.1

1

10

q (Å-1) Fig. 1. SAXS curves for simulated data. The a values were varied, d = 2.5; n = 100; c = 1; r = 5; A = 100; back = 5.

where Imeasured(q) and Imodell(q) indicate the measured and calculated intensities, as well as B is the constant background. The estimated scattering intensities range over several orders of magnitude. The use of logarithmic functions for fitting reduces the values of the intensity to the same level, so that the highest and lowest quantities and so their error bars are commeasurable with each other. Managing so all the measured data points, they have approximately the same weight during the fit procedure.

3. Results 3.1. Evaluation of small angle X-ray scattering data The scattering behaviors were determined over a wide range of q-values, ranging between 0.002 and 1.0 Å1. Thus, the measurements yield structural information from 1 to 2 nm to the micrometer scale. The fractal structure of aluminosilicate aerogel (sample 1) was investigated with the SAXS laboratory equipment over a q-range of 0.014–1.0 Å1 and with the USAXS equipment over a q-range of 0.002–0.03 Å1 (Fig. 2). The figure shows the SAXS and USAXS data in log–log plot and the results of several fitting programs. The parameters obtained from the various fits (powerlaw, Freltoft, modified Freltoft, Emmerling, and Emmerling–Fratzl fits) are listed in Table 1. The SAXS curve is essentially linear on a log–log plot of I versus q, it follows a power-law: I(q)  qD. A small deviation can be observed at high q-range due to the incoherent scattering. The slope (between 2 and 3) of the SAXS curve indicates a mass fractal structure for the aerogel samples. The most precise fitting program proved to be the Freltoft equation modified with a, the least exact was the Emmerling fit (see the root mean square (RMS) error in Fig. 2 displaying the precision of the fits!). Considering the nearly linear SAXS curve over the q-range and the most precise fitting, it can be established that the fractal structure is characteristic in wide range, the elementary units are not three-dimensional particles in this system, and the units have fractal behaviors. The new parameter (a) obtained from the modified Freltoft approximation indicates a fractal-like character of 2.3 dimension for elementary units. A 5-year-old aluminosilicate aerogel (sample 2) was prepared under the same condition as that of sample 1 and kept under ambient temperature and pressure in a closed sample holder. The fractal structure of aerogel sample was degraded in 5 years, the fractal domains have become smaller and more compact. The aged aerogel (sample 2) was analyzed in the q-range of 0.005–0.5 Å1 utilizing the JUSIFA beamline of HASYLAB at DESY (Fig. 3, Table 2). The most important difference on the SAXS curve for aged aerogel (sample 2) to that for the few weeks old aerogel (sample 1) is a deviation from linearity at low q (<0.025 Å1) due to the smaller fractal domain (n  10 nm, Table 2). The SAXS data of the 5-year-old aerogel could very well be fitted to both Freltoft and modified Freltoft approximations. (See the RMS error on Fig. 3!). However, the fit of modified Freltoft expression resulted in an unreal a value (the value of 1.4 would mean 1.2 dimension). Even Emmerling and Emmerling–Fratzl fits were more effectively applied for the evaluation of SAXS data of the 5-year-old aerogel (sample 2) than in the case of few weeks old aerogel (sample 1). The precision of the Freltoft fit and the unreal value of a reveal that the elementary units building the fractal structure are 3D particles with smooth or rough surfaces. Gyration radius obtained from Guinier-plot is not a well defined, reliable parameter in the case of fractal structure without any strongly marked upper-limit length scale for fractal domain. In lack of that, the unsure evaluation in Guinier-plot depends considerably on the q-range (Table 3). The larger the initial value of qrange, the larger the gyration radius is.

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8

8

10

-1

=

-2 .6 5

6

5

10

μ

3

Data: Aluminosilicate 1 Model: Freltoft q = 0,19 Å

10

2

10

0.01

4

-1

q = 0,068 Å

4

10

0 .4 -3

-1

=

Intensity

μ

log Intensity

6

10

-3

10

q = 1.9 10 Å

7

USAXS + SAXS curves Freltoft fit

USAXS + SAXS curves

RMS eror = 0.00135 2

0.1

1

0.01

q (Å-1) 8

USAXS +SAXS curves Emmerling fit

7

6

6

log Intensity

log Intensity

1

8 USAXS + SAXS curves modified Freltoft fit

4

Data: Aluminosilicate 1 Model: modified Freltoft 2

0.1

q (Å-1)

RMS error

5 4 3

= 0.00019

Data: Aluminosilicate 1 Model: Emmerling RMS error

= 0.02613

2 0.01

0.1

1

0.01

0.1

1

q (Å-1)

q (Å-1)

Fig. 2. The fractal structure of aluminosilicate aerogel (sample 1) was measured with SAXS laboratory equipment and with USAXS on the BW4 beamline of HASYLAB at DESY. USAXS and SAXS data were evaluated by Freltoft, modified Freltoft, and by Emmerling approximations (RMS: root mean square error).

Table 1 Characteristic parameters of aerogel fractal structure Measuring method h

USAXS + SAXS h

USAXS + SAXS USAXSh + SAXS USAXSh + SAXS USAXSh + SAXS Nitrogen sorption

Evaluation method a

Power-law (l) Power lawa (l) Freltoftc Mod. Freltoftd Emmerlinge Emmerling–Fratzlf BET

Evaluated q-range (Å1) 2

2

1  10 –5  10 6.0  102–0.2 2  103–1.0 2  103–1.0 2  103–1.0 2  103–1.0

Fractal dimension (d) 2.65 ± 0.01 (3.45 ± 0.01) 2.43 ± 0.02 2.26 ± 0.01 (1.01 ± 0.24) 1.97 ± 0.02 2.59 ± 0.1

330 10.3b 21.8 ± 7 28.7 ± 4 1g – –

a

l: slope of the SAXS curve in log–log plot. Calculated from the q of crossover point (2p/q) on the SAXS curve in log–log plot. SAXS curve was fitted by Freltoft. Modified Freltoft. Emmerling. Emmerling–Fratzl approximations. The size was >1–200 nm. USAXS experiment was performed on the BW4 beamline of HASYLAB at DESY. Calculated from the q (2p/q) of maximum intensity of the SAXS curve in log–log plot.

d e f g h i

Besides the fractal, the aggregate structure is the other characteristic aerogel system. The aggregate structure defines a random packing of colloidal particles. Silica aerogel (sample 3) was prepared under basic condition resulting in aggregate structure (Fig. 4, Table 4). The slope of the SAXS curve in the log–log plot is very close to 4, which indicates a real aggregate structure of particles with smooth surfaces (Fig. 4). In contradiction to the scattering of fractal aerogels, the Guinier-plot was well applied for the SAXS data of silica aggregates yielding a characteristic length of 5.6 nm. The parameters derived from several fits to the SAXS data varied in wide range and presented unreal material information (Table 4).

Primary particle radius (r) (nm)

i

b c

Fractal correlation length (n) (nm)

3.3b 1.8 ± 0.2 3.0 ± 0.06 8.1 ± 1.0 4.2 ± 0.11 –

3.2. Transmission electron micrographs Transmission electron micrographs show the size of elementary units, fractal domains, and aggregates (Fig. 5). The size distributions detected by TEM are represented in Fig. 6. The radii of the elementary units building the fractal structure of aluminosilicate aerogel (sample 1) observed on the TEM image are around 3 nm. This value corresponds to the primary particle radius obtained by modified Freltoft approximation (r = 3.0–3.2 nm) from USAXS and SAXS data. TEM photograph of aged aerogel (sample 2) reveals fractal domains of 9–10 nm and primary particles of 1–1.5 nm sup-

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7

10

6

10

5

10

4

10

3

3

10

2

2

SAXS curve measured by JUSIF A synchrotron beamline

μ

=

-3

.1

8

SAXS curve measured by JUSIF A synchrotron beamline Freltoft fit

6

log Intensity

Intensity

7

10

5

4

Data: Aged Aluminosilicate Model: Freltoft RMS error

0.01

= 0.00011

0.01

0.1

0.1

q(Å-1) 7

q(Å-1) 7

SAXS curve measured by JUSIFA synchrotron beamline modified Freltoft fit

6

log Intensity

log Intensity

6

SAXS curve measured by JUSIFAsynchrotron beamline Emmerling fit

5

4

Data: Aged aluminosilicate Model: modified Freltoft

3

RMS error

5

4

3

Data: Aged aluminosilicate Model: Emmerling RMS error

= 0.00008

2

= 0.00894

2 0.01

0.1

0.01

q(Å-1)

0.1

q(Å-1)

Fig. 3. Aged (5-year-old) aluminosilicate aerogel (sample 2) was investigated with the JUSIFA beamline of HASYLAB at DESY. SAXS data were fitted by Freltoft, modified Freltoft, and by Emmerling approximations.

Table 2 Characteristic parameters of 5-year-old aerogel fractal structure Measuring method

Evaluation method

q-range (Å1)

Fractal dimension (d)

Fractal correlation length (n) (nm)

Primary particle radius (r) (nm)

SAXS/synchrotrona SAXS/synchrotrona SAXS/synchrotrona SAXS/synchrotrona SAXS/synchrotrona

Power lawb (l) Freltoftd Mod. Freltofte Emmerlingf Emmerling–Fratzlg

5  102–0.34 5  103–0.5 5  103–0.5 5  103–0.5 5  103–0.5

(3.18 ± 0.01) 2.54 ± 0.02 2.22 ± 0.07 (1.04 ± 0.40) 2.00 ± 0.03

– 9.0 ± 0.15 11.7 ± 0.7 1h –

– 1.5 ± 0.08 5.2 ± 0.3 7.1 ± 1.1 3.35 ± 0.2

a

Five-year-old aluminosilicate aerogel (sample 2) was measured with the JUSIFA beamline of HASYLAB at DESY. l: slope of the SAXS curve in log–log plot. d,e,f,g,h See at the Table 1. b

4. Discussion Table 3 Guinier-evaluation Measuring method

Evaluation method

q-range (Å1)

Gyration radius (nm)

SAXSa SAXS/synchrotronb USAXSc + SAXS

Guinier Guinier Guinier

1.4  102–1.0 5  103–0.5 2  103–1.0

9.0 ± 0.10 12.0 ± 0.15 35.4 ± 0.20

a

Aluminosilicate aerogel (sample 1) was measured with laboratory equipment. SAXS measurements were carried out at the JUSIFA beamline of HASYLAB at DESY. c USAXS experiment was performed with the BW4 beamline of HASYLAB at DESY. b

porting the Freltoft evaluation model (n = 9 and r = 1.5 nm). Sizes (radii) of around 5 nm can be measured on the TEM image of silica aggregates (sample 3), which approach the gyration radius of 5.6 nm. The SAXS measurements did not detect any changes in the aerogel structures after the TEM studies.

This work mainly addresses the characterization of several evaluation methods of scattering data to determine the nanostructure of amorphous aerogels. Three kinds of aerogel structure provided the samples of these investigations; the fractal of wide range, the fractal of limited size, and the aggregate system. The fractal of wide range denotes a structure built up from elementary units having fractallike character; the fractal of limited size signifies a network of 3D compact primary units. The size of the primary particles and their fractal or aggregate clusters ranges from a few to a few hundreds of Angstrom thus they represent a suitable object for SAXS studies. 4.1. Characterizing the evaluation programs of the SAXS data Besides the 50 years old Guinier-formula, there are two newly developed equations for fitting the SAXS curves of fractal objects: the Freltoft (Eq. (6)) [2] and the Emmerling formulas (Eq. (8)) [4].

5471

6

10

5

10

4

10

3

10

2

6

log Intensity

10

USAXS + SAXS curves Freltoft fit

7

USAXS + SAXS curves q = 0.014 Å-1

7

97 -3.

10

μ=

Intensity

K. Sinkó et al. / Journal of Non-Crystalline Solids 354 (2008) 5466–5474

5 4 3

Data: Silica Model: Freltoft

2

RMS error

0.01

0.1

1 1E-3

1

= 0.01325

0.01

q (Å-1) USAXS +SAXS curves modified Freltoft fit

7

1

USAXS + SAXS curves Emmerling fit

7

6

6

log Intensity

log Intensity

0.1

q (Å-1)

5 4 3

Data: Silica Model: modified Freltoft

2

RMS error

1E-3

= 0.0125

0.01

0.1

1

q (Å-1)

5 4 3

Data: Silica Model: Emmerling fit

2

RMS error

1E-3

= 0.01508 0.01

0.1

1

q (Å-1)

Fig. 4. The aggregate structure of silica aerogel (sample 3) was studied with SAXS laboratory equipment and with USAXS on the BW4 beamline of HASYLAB at DESY. USAXS and SAXS data were fitted by Emmerling and modified Freltoft expressions.

Table 4 Characteristic parameters of aggregate aerogel structure Measuring method

Evaluation method

Evaluated q-range (Å1)

Fractal dimension (d)

Fractal correlation length (n) (nm)

Primary particle radius (r) (nm)

SAXSa

Power lawb (l) Freltoftc

0.045–0.34

(3.99 ± 0.01)

–

–

2.98 ± 0.06

4.4 ± 0.4

5.6 ± 0.2

2.12 ± 1.4

7.4 ± 10.0

5.6 ± 3.0

22 ± 85.0

7.4 ± 1.6

SAXS

a

SAXSa SAXSa

Mod. Freltoftd Emmerlinge

2

1.4  10 – 1.0 1.4  102– 1.0 1.4  102– 1.0

(1.0 ± 0.04)

SAXS curve was fitted by Freltoftc, modified Freltoftd, Emmerlinge, approximations. a Silicate aerogel (sample 3) was measured with laboratory equipment. b l: slope of the SAXS curve in log–log plot.

The fitting program developed by Freltoft et al. for the evaluation of the scattered intensity to the fractal structure of the aggregates of limited size calculates with six parameters (Eqs. (4)–(6)) [2]. Freltoft et al. introduced a new fractal parameter (n) and established the usefulness of the concept of an upper length-scale cutoff n, which must be used to analyze scattering data from fractal objects if the aggregated clusters have a finite size or become entangled, thus losing the power-law correlations which exist on a single cluster [2]. Evaluations of the present SAXS data proved that the Freltoft approximation can be effectively used for the scattering data of fractal structure if the elementary units building the

fractal domain are 3D particles with smooth or slightly rough surfaces and the sizes of fractal object are inside the q-range of scattering (see the SAXS curve for 5-year-old aluminosilicate aerogel, sample 2 in Fig. 3!). TEM photography verified the parameters (r, n) obtained by Freltoft fit in the evaluation of SAXS data for fractal system of limited size. If the elementary units show fractal behavior, the Freltoft fit does not find the ‘‘knee” of the size of elementary units on the SAXS curve (for example the Freltoft fit did not find the size of 3.2 nm in the case of sample 1.). The modified Freltoft expression (Eqs. (11) and (12)) is the most suitable for the evaluation of the SAXS data for fractal structure of wide range. The suitability of the modified expression has been proved by not only its accuracy but also by its demonstrably good results. The TEM micrographs verified the size of elementary units obtained by modified Freltoft fit in the case of fractal systems of wide range. The newly introduced parameter (a) provides the representation of the structural density of the elementary units. The Emmerling function contains one less parameters compared to that of Freltoft approximation [4]. That has been caused by normalization during the calculation. The fact that the parameter C in the Freltoft formula is fixed by the values of fractal dimension (D) and particle radius (R) and both parameters are mainly determined by certain parts of the measured curve, fixes the ratio of the structure and the particle form factors in the end-formula, which influences in most cases the quality of the fit. Comparing the Freltoft and Emmerling equations, the principal difference can be found in these parts; in the Freltoft formula there is

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Fig. 5. TEM photographs of fresh (A, sample 1), aged (B, sample 2) aluminosilicate, and silica aerogels (C, sample 3) showing their elementary units.

The Emmerling and the Freltoft formulae are mathematically equivalent if

Aluminosilicate aerogel Silica aerogel

90

number of particles

80

C¼

70 60 50 40 30 20 10 0 3

4

5

6

7

8

9

10

11

12

13

14

size of particles / nm Fig. 6. Size distribution of elementary units of aluminosilicate (sample 1) and silica (sample 3) aerogels detected by TEM.

X = [1 + (q  n)2] and the Emmerling formula contains Y = [1 + 1/ (q  n)2]. The complete Freltoft formula is:

!2 pffiffiffi 2 2 2 IðqÞ ¼ back þ A 1 þ q R 3 ! C  nD 2 2 1D 2 ð1 þ q n Þ  CðD  1Þ sin½ðD  1ÞarctgðqnÞ :  1þ qn ð14Þ The complete Emmerling formula:

!2 0 !1D pffiffiffi 2 DCðD  1Þ 1 2 2 2 @ IðqÞ ¼ back þ A 1 þ 1þ 1 þ q R 3 ðqRÞD ðqnÞ2  sin½ðD  1ÞarctgðqnÞÞ:

ð15Þ

D RD

:

ð16Þ

In the case of scattering curves for real aerogel systems, several differences must be pointed out. If the correlation length (n) reaches the border of the measure range and Y is close to one, then n has to be hardly fitted. On the other hand, in the same case X has been determined by the value of n. This fact makes n more determined giving more stability to the fitted function. Thus the Emmerling approximation can be fitted to scattering data for fractal aerogel with less result than the Freltoft fit (see Figs. 2 and 3!). The Emmerling formula modified by n = 1 (Eq. (15)) can be more effectively applied for the evaluation of the scattering data for fractal structure of not limited size [6] (see Table 1!). The values of the fractal dimension and the size of elementary units are more real and approach the mean values. The representation of the data in a Guinier-plot clearly indicates that there is no simple characteristic length scale to be deduced from the data of fractal systems (Table 3). If the elementary units building the aggregates are 3D particles with smooth surface and the slope of the SAXS curve in log–log plot is near 4, Guinier-plot is worth using for the evaluation of the scattering data. 4.2. Characterization of the structure of aerogels On the basis of Freltoft fit modified with a parameter, the aluminosilicate aerogel samples (sample 1) have mass fractal network of about 2.4 dimension, the fractal objects spread from a few Angstrom to about 60 nm. The mean radius of elementary units building up the fractal domain is about 3 nm (Table 1, data of modified Freltoft fit and TEM image A). Considering the value of du (2.3) calculated from a, the elementary units are not compact particles but an open randomly branched system quantified by about 2.3 fractal dimension. This value is very similar to the mass fractal dimension

K. Sinkó et al. / Journal of Non-Crystalline Solids 354 (2008) 5466–5474

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Fig. 7. Bond system of the fractal-like elementary units those built up the aluminosilicate network (sample 1) prepared from Al nitrate and TEOS. All geometries were performed using Gauss View. The size of elementary unit controlled by Gauss View is 3 nm.

of whole fractal object verifying the fractal characters in wide range. Fig. 7 symbolises the chemical bond system of fractal-like elementary units (3 nm). All geometries were performed using Gauss View [19] and considering the Loewenstein‘ rule [20], Al– O–Al bonds are not preferred; the previous results of Al MAS NMR and FT-IR [21]; and the values for Al–O–Si, Si–O–Si bond angles [22–24] and bond length taken from Ref. [22,25].

A 5-year-old aluminosilicate aerogel (sample 2) proved to be also a fractal system of 2.5 mass fractal dimension, the aging did not lead to dramatic changes. During aging, the fractal domains were slightly degraded to smaller units of 9 nm and their elementary units became more compact (Table 2, data of Freltoft fit and TEM image B). The 3D elementary units can be characterized by radii of 1–1.5 nm and rough surfaces. Fig. 8 depicts the chemical bond system of 3D elementary units (1.5 nm). The other characteristic structure of aerogels, the aggregate system (sample 3) denotes a random building of colloidal particles. The silica aerogels (sample 3) were prepared under basic conditions. The base catalysis leads to relatively rapid condensation rate and smaller hydrolysis rate resulting in an aggregate system. The slope (4) of the SAXS curve in log–log plot indicates the aggregate structure built up from 3D particles with smooth surfaces. The result of Guinier-plot provides the aggregate structure with a characteristic length of 5.6 nm. The fitting program applied in this work yielded not real material parameters (Table 4). The TEM image proves an average size of 5 nm in the aggregate structure. 5. Conclusions

Fig. 8. Bond system of the 3D elementary units those built up the aged aluminosilicate network (sample 2) prepared from Al nitrate and TEOS. All geometries were performed using Gauss View. The size of elementary unit controlled by Gauss View is 1.5 nm.

The characterization of the structure investigation techniques and their data evaluation programs were focused on the nano sizes (from a few to a few hundreds of Angstrom) of porous systems. The measurements (SAXS, USAXS, TEM, and SEM) were performed on fractal structures of aluminosilicate aerogels and aggregate systems of silica aerogel. The comparison of the evaluation programs occurred with respect to the precision of the fitting and the obtained structural parameters. TEM photography verified the parameters resulted by the most suitable evaluation program.

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The evaluation of the scattering data depends strongly on the aerogel structures. The Freltoft approximation can be effectively used for the evaluation of the scattering data for fractal structure of limited size. Thus it is necessary that the elementary units to be 3D particles with smooth or slightly rough surfaces. The Emmerling approximation can be fitted to the scattering data for fractal aerogel with less result than the Freltoft fit. The performance of Emmerling fit can be improved by using an infinite size for fractal object. If the primary elementary units exhibit fractal behaviors, we suggest the introduction of a new parameter (a) in the form factor of Freltoft expression, which parameter provides the representation of the structural properties of elementary units. The SAXS data for aggregate structure of 3D elementary particles with smooth surfaces can be evaluated with the simple powerlaw expression in the Porod’s region and with the Guinier-plot. Acknowledgment We thank Herwig Peterlik and Nicola Hüsing for all their help. This study has been supported by OTKA NK 68750 funds, and I20060083 EC in HASYLAB DESY. References [1] A. Guinier, G. Fournet, Small-Angle Scattering of X-ray, John Wiley, New York, 1955.

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