Advances in Colloid and Interface Science 76]77 Ž1998. 57]69
Structure of Ru]Ti oxide aerogels: a SANS study Celia I. Merzbacher a,U , John G. Barker b, Karen E. Swider c , Debra R. Rolisonc a
b
Optical Sciences Di¨ ision, Na¨ al Research Laboratory, Washington, DC 20375, USA Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA c Chemistry Di¨ ision, Na¨ al Research Laboratory, Washington, DC 20375, USA
Abstract Small-angle neutron scattering ŽSANS. has been used to characterize Ru]Ti oxide aerogels annealed under various atmospheres and temperatures. The pores were filled with H 2 OrD 2 O solutions that match the scattering properties of either RuO 2 or TiO 2 , therefore scattering from the remaining unmatched phase is measured independently. The structure of the RuO 2 and TiO 2 components of the aerogel detected by SANS depends more on annealing temperature than annealing atmosphere. The results are consistent with a branched, polymeric network made up of 8]16 nm TiO 2 particles and - 5-nm RuO 2 particles. The SANS results are supported by TEM. This study demonstrates that contrastmatching SANS is complementary to other techniques for characterizing the structure of two-phase aerogels. Published by Elsevier Science B.V. Keywords: Aerogel; SANS; Contrast matching; Ruthenium]titanium oxide; Nanoscale structure
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . 2. Experimental procedure . . . . . . . . . . . . . 2.1. Sample preparation and characterization 2.2. SANS experiments . . . . . . . . . . . . . . U
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Corresponding author. Tel.: q1 202 404 7987; fax: q1 202 404 8114; e-mail:
[email protected]
0001-8686r98r$19.00 Published by Elsevier Science B.V. PII S0001-8686Ž98.00041-4
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3. Results and discussion 4. Summary . . . . . . . . . Acknowledgments . . . . . References . . . . . . . . .
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1. Introduction Electrically conductive ceramic compounds based on mixtures of ruthenium dioxide and titanium dioxide are important electrode materials for electrocatalytic evolution of chlorine, oxygen and hydrogen w1x. In these Ru]Ti oxide electrodes, electrocatalysis occurs primarily at the electrically conductive RuO 2 surface. Insulating TiO 2 , in the isostructural rutile form, is added to stabilize the RuO 2 and decrease the cost of the electrodes. The reaction rates of the bulk ceramic anodes have been substantially improved by increasing the surface area of the RuO 2 w1x. To further enhance these improvements, Ru]Ti oxide aerogels have been developed w2x. Although many aerogel compositions have been investigated for catalytic applications w3x, this is the first known ruthenium-bearing example. Aerogels are attractive for catalysis because of their low density and high surface area. Furthermore, some inorganic oxide aerogels can withstand heating to at least 5008C without collapse or shrinkage. In order to achieve these properties, aerogels are typically prepared by synthesizing a sol-gel from metal alkoxide precursors in an excess of solvent. The solvent is then removed from the pores under supercritical conditions, thereby avoiding the large capillary forces and shrinkage that occurs when liquid is removed from small pores. This process has been used to make silica aerogels with densities as low as 0.003 grcm3 and surface areas of up to 1000 m2rg w4x. The structure of aerogels has been determined primarily by transmission electron microscopy w5,6x and small angle scattering w7]10x. These studies show that, in general, aerogels are composed of microporous particles on the order of 10 nm in diameter, connected in an open network that encloses a large volume fraction of mesopores Ž2]50 nm diameter.. In the Ru]Ti oxide aerogels, we are particularly interested in obtaining information on the structure of the Ru oxide phase. As long as a fully connected three-dimensional network of electrically conducting RuO 2 is present, ruthenia-like electrical properties are maintained. In the bulk mixed oxides, materials prepared with up to 70% TiO 2 exhibit the electronic conductivity of RuO 2 w11x. The electrical properties of the Ru]Ti oxide aerogels, as determined by impedance spectroscopy, are strongly dependent on the atmosphere and temperature of the final heat treatment, or annealing w2x. In the conductive samples, the conductivity is dominated by protonic conduction through a hydrous surface layer. This is in contrast to the model of conductivity in bulk electrodes, in which conduction occurs through an interconnected RuO 2 phase w1x. The electrical properties of the high surface area Ru]Ti oxide aerogels are dominated by their
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interfacial or surface conductivity. Information about the aerogel RuO 2 network beneath the surfaces, however, was not revealed by the impedance characterization. Contrast-matching SANS has been used in this study to independently characterize the structure of the Ru and Ti oxide portions of mixed-oxide aerogels. Small angle neutron scattering ŽSANS. is ideal for the characterization of aerogels because it probes structures on the scale of 1]100 nm. The observed scattered intensity depends on the contrast in the scattering length density, r l , of the different phases in the sample. The scattering length density is the atomic coherent scattering length, b, normalized on a volume element basis:
rl s
S bi V
Ž1.
where the numerator is the weighted sum of the coherent scattering lengths for the individual atoms in the molecule and V is the molecular volume w12x. The value of b is a function of isotopic number and nuclear spin state, and does not vary regularly with atomic number Žsee tabulated values in Ref. w13x.. In fact, hydrogen has a low coherent scattering length, whereas deuterium has a very high value. Therefore, the values of r l for H 2 O and D 2 O are relatively low and high, respectively. It is thus possible to match the r l of most compounds by an appropriate mixture of H 2 O and D 2 O. ‘Contrast matching’ can be used to characterize porous two-phase materials. The pores are filled with an H 2 O:D 2 O mixture that matches the scattering length density of one phase, allowing characterization of the non-contrast-matched phase. This technique has been particularly successful, for example, in the study of polymer]polymer blends w14x. Interpretation of small-angle scattering by single-phase aerogels, polymers and colloids in terms of fractal dimension has been the subject of various investigations w15]17x. If a structure is a mass fractal, mass, M, is related to radius, R, by the mass fractal dimension, Dm : M A R Dm .
Ž2.
If a structure is dense in its bulk, that is Dm s 3, but has a fractally rough surface, the surface area, SA, is related to the radius, R, by the surface fractal dimension, Ds : SA A R D s .
Ž3.
Both the surface and mass fractal dimensions can be related to the slope of a logŽ q . vs. logŽ I ., or Porod, plot. In the case of mass fractals I Ž q . A qyD m .
Ž4.
For surface fractals, it has been shown w18x that I Ž q . A q D sy6 .
Ž5.
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C.I. Merzbacher et al. r Ad¨ . Colloid Interface Sci. 76]77 (1998) 57]69
In general, scattering curves from polymeric mass fractal structures have slopes of y1 to y3 and rough colloids give slopes of y3 to y4 w15x. Scattering from non-fractal, smooth particles yields a slope of y4. By way of example, a typical spectrum for silica aerogel, which is the most widely studied aerogel composition, is shown in Fig. 1. The interpretation of this curve, as first published by Schaefer and Keefer w7x, is as follows: at short lengths Žhigh q . scattering from smooth particles results in a slope of y4, at intermediate lengths scattering is from a mass fractal structure with dimension of 2, and at long lengths the structure is uniform and scattering is independent of q Žslope s 0.. This type of interpretation should be applicable to the spectra from aerogels filled with TiO 2- and RuO 2-matching liquid, to describe the RuO 2 and TiO 2 components independently. Strictly speaking, a fractal is self-similar over several orders of magnitude, and therefore the Porod slopes should also be linear over a wide range of q values. This is not always the case for the Ru]Ti oxide aerogels. Rather than conclude that these materials are fractal in nature, we shall simply draw parallels between our data and previous results for true fractals.
2. Experimental procedure 2.1. Sample preparation and characterization Ru]Ti oxide aerogels were prepared by a method that is described in detail elsewhere w2x. Briefly, titaniumŽIV.isopropoxide, TiŽO i Pr.4 ŽAldrich., and anhydrous RuCl 3 ŽAlfa. are used as the metal precursors. Because no fully substituted Ru metal alkoxide is commercially available, a precursor solution was prepared by refluxing 9.1 wt.% RuCl 3 in ethanol. Supernatant from this solution was mixed
Fig. 1. Typical small-angle scattering spectrum of a silica aerogel Žafter Ref. w7x..
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with concentrated HNO3 and distilled, deionized water. This mixture was added to a mixture of equal parts TiŽO i Pr.4 and ethanol. The quantities of each component were calculated such that the final volume was approximately 50 ml and the ratio ŽRu q Ti.:H 2 O:Hq:EtOH was 1:6:0.08:20. The Ru]Ti sol was immediately poured into 5-ml polypropylene molds and gelled in ; 3 min. After aging for 2.5 days, the gels were removed from the molds and washed in acetone four times over 2 days and then dried under supercritical CO 2 . After drying, the samples were annealed in order to remove residual chloride and organic phases. Thermal analyses were performed using a TA Instruments Model 951. The temperature at which these phases are completely volatilized ŽTR . depends on the annealing atmosphere as shown in Fig. 2. TR under Ar, TRŽAr., is 4108C whereas under oxygen, TRŽO., the volatiles are completely removed at 3208C. The electrical conductivity, as measured by impedance spectroscopy, was maximum in samples annealed close to TR and decreased as the annealing temperature was raised. Samples were annealed at and above TR under flowing ‘wet Ar’ Ži.e. Ar bubbled through water. or ‘wet O 2 ’. After annealing, the Ru]Ti oxide aerogel monoliths were cracked. All scattering experiments were carried out on powdered material, which was prepared by grinding the aerogel monoliths using an agate mortar and pestle for ; 15 s. The bulk composition of samples prepared by this procedure is Ru 0.32Ti 0.68 O 2 based on atomic absorption spectroscopy ŽCorning Analytical Laboratory.. The bulk density of the annealed monolithic aerogels measured using a Moore]Van Slyke specific gravity bottle is ; 12% of the theoretical density of a 0.32:0.68 mixture of crystalline RuO 2 :TiO 2 .
Fig. 2. Thermogravimetric analysis of Ru]Ti oxide aerogel samples under flowing wet Ar Žsolid. and wet O 2 Ždashed..
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The presence of crystalline phases was determined by powder X-ray diffraction ŽXRD. ŽScintag XDS2000.. Surface areas were measured by nitrogen adsorption using the single-point BET method ŽQuantachrome Monosorb.. Transmission electron microscopy of a sample that had been annealed under wet Ar at TRŽAr. Ž4108C. was performed using a Phillips CM30 TEM operating at 100 kV. Aerogel powders were captured on holey carbon grids. Efforts to observe electron diffraction from crystalline portions of the sample were not successful. 2.2. SANS experiments Scattering data were collected on a 30-m SANS spectrometer at the National Institute of Standards and Technology ŽNIST. ŽGaithersburg, MD.. A neutron wavelength of 0.5 nm and two sample-to-detector distances, 2 and 11 m, were used to cover a momentum transfer Ž q . range of 0.04]3.5 nmy1 . The raw data Žscattered intensity, I, as a function of q . were corrected for background and empty cell scattering and then converted to values of differential scattering cross section using the SANS data reduction software developed at NIST w19x. The flat incoherent background, which was subtracted, was determined for each sample from the slope of the high-q data in a plot of q m = I Ž q . vs. q m , where m is the power law exponent relating q and I Ž q . at high q. Samples were run as dry powders and with either RuO 2- or TiO 2-contrast matching H 2 OrD 2 O mixtures filling the pores. The powders were loaded and weighed in 1-mm thick cells with quartz windows. The spectrum of each dry sample was collected and then the cell was filled with the appropriate H 2 OrD 2 O mixture. The scattering length densities for the phases of interest and matching H 2 OrD 2 O ratios are shown in Table 1. The mass densities for RuO 2 and TiO 2 Žshown in Table 1. were used to calculate the scattering lengths of the aerogel phases to be matched. The samples were placed under slightly reduced pressure in order to remove larger bubbles from the cell. A comparison of the SANS spectra of a silica aerogel monolith vs. powder was performed and showed that there is no effect due to the powdering process on the scattered intensity in the q range studied. In addition, the ability to completely refill all pores that are large enough to cause scattering ŽR 2 nm. has been confirmed using a monolithic silica aerogel w20x and a titania aerogel powder
Table 1 Mass density, calculated scattering length density and matching H 2 O:D 2 O ratio for RuO 2 and TiO 2 Compound
Mass density Žgrcm3 .
Scattering length density Ž=1010 cmy2 .
Matching H2 O:D2 O ratio
RuO2 TiO2 H2 O D2 O
6.97 4.26 1.0 1.1
5.94 2.67 y0.56 6.3
5:95 53:47 } }
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prepared by the procedure described by Dagan and Tomkiewicz w21x. In these studies, samples refilled with silica- or titania-matching H 2 OrD 2 O liquids, respectively, showed no scattering from the mesopores. The potential deformation of the aerogel structure by the rewetting process has also been investigated by comparing the SANS of silica aerogel with air vs. D 2 O in the pores. The absolute intensity is slightly different due to the different contrast between air and SiO 2 vs. D 2 O and SiO 2 , but the lineshapes are identical, indicating that there is no permanent effect on the silica aerogel structure due to rewetting. Prior to annealing, the Ru]Ti oxide aerogels are amorphous, but after annealing to G 3208C, XRD indicates the presence of rutile RuO 2 and anatase TiO 2 w2x. No ternary Ru]Ti oxide phase or solid solution of Ru into TiO 2 or vice versa was observed at the level detectable by XRD Ž5]10 mol %.. At annealing temperatures G 6008C, rutile TiO 2 increases at the expense of anatase. Although XRD does not preclude the presence of a non-crystalline component, no evidence of crystallization upon heating to 5008C in air or Ar was observed by differential scanning calorimetry, which suggests that an amorphous phase is not a significant component. Therefore, for ease of discussion, we refer to the unmatched fraction as ‘TiO 2 ’ in the samples filled with RuO 2-matching liquid and vice versa. 3. Results and discussion Porod plots of the background-corrected SANS spectra of annealed Ru]Ti oxide aerogels with RuO 2- and TiO 2-matching D 2 OrH 2 O solutions in the pores are shown in Figs. 3 and 4, respectively. Distinctly different scattering is observed from the TiO 2 and RuO 2 components of the aerogels for a given annealing atmosphere and temperature. Fig. 5 shows TiO 2-matched SANS curves for a series of samples annealed in wet Ar at temperatures up to 8008C. The Bragg relation, q s 2prL, allows correlation lengths Ž L. to be associated with features in the SANS spectra. The structure of the TiO 2 component of the Ru]Ti oxide aerogels is revealed by the scattering from samples filled with RuO 2-matching liquid ŽFig. 3.. All of the curves consist of three linear segments. At short length scales Žhigh q ., the Porod slope is y3.3 to y3.9, which is consistent with rough colloidal particles. At longer lengths the slopes of y1.4 to y1.9 are attributed to a more polymeric, or branched, structure built from the colloidal particles. The crossover between these two regimes increases from 8 to 16 nm with an increase in annealing temperature from 320 to 4608C. This correlates to the gradual coarsening of the primary particles as a function of annealing temperature. The more negative slopes Žy2.7. at longer lengths Žlow q . are due to scattering from the powder particles. They are not due to trapped bubbles, because the smooth surfaces of bubbles would lead to a slope of y4. The structure of the RuO 2 component of the Ru]Ti oxide aerogels is inferred from the scattering observed from samples filled with TiO 2-matching liquid ŽFigs. 4 and 5.. The TiO 2-matching liquid has a greater H 2 O:D 2 O ratio than the RuO 2matching liquid, and therefore the spectra of the TiO 2-matched samples have a
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C.I. Merzbacher et al. r Ad¨ . Colloid Interface Sci. 76]77 (1998) 57]69
Fig. 3. Small-angle neutron scattering curves of Ru]Ti oxide aerogels with pores filled by RuO 2-matching liquid. Annealing atmosphere and temperatures are indicated for each spectrum. Slopes were determined by least squares fitting. Errors are smaller than the size of the symbol.
higher proton-induced incoherent background. This effectively obscures the highest q scattering from the sample and reduces the q range of the data compared to the RuO 2-matched spectra. Annealing at temperatures up to 4108C, irrespective of atmosphere, yields curves with a uniform slope of approximately y3. Porod slopes of y3 can result from scattering by very rough colloid particles or relatively dense polymeric structures. Because the linear region extends to lengths of ) 100 nm, the former explanation can be ruled out based on the TEM of a sample annealed under wet Ar at 4108C ŽFig. 6.. The image clearly shows ; 10-nm, not 100-nm, particles connected in a porous network. The Porod plots for samples annealed in wet Ar at 50 and 1908C above TRŽAr. Ž460 and 6008C. have a slope of y1.4 at short length scales ŽFig. 5.. This slope is similar to that observed at somewhat longer length scales from the TiO 2 compo-
C.I. Merzbacher et al. r Ad¨ . Colloid Interface Sci. 76]77 (1998) 57]69
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Fig. 4. Small-angle neutron scattering curves of Ru]Ti oxide aerogels with pores filled by TiO 2-matching liquid. Annealing atmosphere and temperatures are indicated for each spectrum. Slopes were determined by least squares fitting. Errors are smaller than the size of the symbol.
nent ŽFig. 3., and may be due to a polymeric network of ruthenia particles that are smaller than ; 5 nm. That is, if data could be collected at higher q, the slopes would become steeper. This interpretation is consistent with the data for the sample annealed at 8008C, which does show a high-q region with a more negative slope. As observed in Fig. 3, spectral features tend to shift to longer lengths Žlower q . with increased annealing temperature as the structure coarsens. Annealing at 8008C would be expected to significantly increase the size of the particles formed initially by the low temperature sol-gel process. The crossover between the two regimes for the sample annealed at 8008C indicates a particle size of 20 nm. As in the spectra from the RuO 2-matched samples, the low-q scattering is believed to be affected by scattering from the powder particles.
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Fig. 5. Small-angle neutron scattering curves of Ru]Ti oxide aerogels with pores filled by TiO 2-matching liquid. All samples were annealed under wet Ar at the temperatures indicated. Slopes were determined by least squares fitting. Errors are smaller than the size of the symbol.
The SANS spectra of samples annealed under wet Ar at TRŽAr. q1908C Ž6008C. and TR q3908C Ž8008C. are shown in Fig. 5 along with the data for samples annealed under the same atmosphere at lower temperatures. The spectrum of the sample annealed at 6008C resembles that of the sample annealed at TRŽAr. q508C Ž4608C.. The sample annealed at 8008C has a spectrum that looks more like the scattering from the RuO 2-matched samples ŽFig. 3. with three distinct linear segments. Compared to the scattering from the RuO 2-matched samples, however, the high-q crossover occurs at lower q Žequivalent to 20 nm., indicating considerable coarsening. Does the structure of the Ru]Ti oxide aerogels inferred from these SANS results correlate to their electrical conductivity? As reported elsewhere w2x, the
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Fig. 6. Bright-field TEM image of Ru]Ti oxide aerogel annealed under wet Ar at 4108C.
electrical properties of these aerogels are a strong function of the annealing conditions. Optimal conductivity was associated with heating under wet Ar to TRŽAr. Ž4108C. and cooling under O 2 . Heating to temperatures above TR significantly reduced the electrical conductivity. For the SANS study, we chose to study separately the effects of annealing atmosphere and temperature. The results indicate that annealing temperature is more important than annealing atmosphere in determining the structure detected by SANS. However, no correlation was observed between the structural variation measured by SANS and the electrical properties. A model has been developed that is consistent with the SANS data, XRD data, and the electrical measurements. TiO 2 sols gel more rapidly than RuO 2 sols during the synthesis of this mixed-oxide material, forming a network of ; 10-nm TiO 2 particles. The RuO 2 sol forms smaller, phase-segregated RuO 2 particles and may also gel on the surfaces of the TiO 2 particles. During annealing in the wet atmosphere, hydrous Ru oxide forms a layer on the RuO 2 , which acts as a protonic conductive pathway. Future SANS experiments of aerogels with coatings deposited on the internal surfaces of single-phase aerogel substrates will give a more complete understanding of the structure in the Ru]Ti oxide aerogels.
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4. Summary Filling Ru]Ti oxide aerogels with either RuO 2- or TiO 2-matching liquid reveals two distinct structures for samples annealed at given conditions. Whereas the electrical properties and temperature at which residual volatiles are removed are strongly dependent on the atmosphere, the average structure of the RuO 2 ŽTiO 2matched. and TiO 2 ŽRuO 2-matched. components of the aerogel detected by SANS is more affected by annealing temperature. The structure appears in general to consist of rough colloidal particles that are connected in a polymeric network. The size of the TiO 2 particles is 8]16 nm, for samples annealed at 320]4608C. The RuO 2 particles appear to be smaller than the detectable limit Ž; 5 nm., except in the sample annealed at 8008C where they have a diameter of 20 nm. These results are consistent with the features observed by TEM. The ability to perform contrast-matching SANS on multi-phase aerogels has been demonstrated in this study. Pores with diameters as small as ; 2 nm can be readily refilled with water. Further characterization of the nanoscale components of the aerogels, in particular the potentially significant, but uncharacterized, hydrous component, will aid in the full understanding of the structure ]property relations in these types of materials.
Acknowledgements Catherine Cotell ŽNaval Research Laboratory. is gratefully acknowledged for performing the TEM analysis. This work was supported by the ONR and DARPA and is based upon activities supported by the National Science Foundation under Agreement No. DMR-9423101. The thoughtful comments of Andrew Allen, Susan Krueger, and an anonymous reviewer improved the paper significantly. Certain commercial equipment and materials are identified in this paper in order to specify experimental procedures adequately. Such identification is not intended to imply recommendation or endorsement of these products.
References w1x w2x w3x w4x w5x w6x w7x
S. Trasatti, Electrochim. Acta 36 Ž1991. 225. K.E. Swider, C.I. Merzbacher, P.L. Hagans, D.R. Rolison, Chem. Mater. 9 Ž1997. 1248. G.M. Pajonk, App. Catal. 72 Ž1991. 217. L.W. Hrubesh, T.M. Tillotson and J.F. Poco, Better Ceramics Through Chemistry IV, in: B.J.J. Zelinski, C.J. Brinker, D.E. Clark, D. R. Ulrich ŽEds.., Materials Research Society Symposium Proceedings, Vol. 180, Materials Research Society, Pittsburgh, PA, 1990, p. 315. P.H. Tewari, A.J. Hunt, J.G. Lieber and K. Lofftus, ‘‘Aerogels’’ , in: J. Fricke ŽEd.., Sprint Proceedings in Physics, Vol. 6, Springer]Verlag, New York, 1986, p. 142. Z. Zhu, L.Y. Tsung, M. Tomkiewicz, Phys. Chem. 99 Ž1995. 15945. D.W. Schaefer, K.D. Keefer, Phys. Rev. Lett. 56 Ž1986. 2199.
C.I. Merzbacher et al. r Ad¨ . Colloid Interface Sci. 76]77 (1998) 57]69
69
w8x A. Emmerling, J. Gross, R. Gerlach, R. Goswin, G. Reichenauer, J. Fricke, H.-G. Haubold, J. Non-cryst. Solids 125 Ž1990. 230. w9x J. Pelous, M. Foret, R. Vacher, J. Non-cryst. Solids 145 Ž1992. 63. w10x Z. Zhu, M. Lin, G. Dagan, M. Tomkiewicz, J. Phys. Chem. 99 Ž1995. 15950. w11x W.A. Gerrard, B.C.H. Steele, J. Appl. Electrochem. 8 Ž1978. 417. w12x O. Glatter and O. Krattky, Small Angle X-ray Scattering. Academic Press, New York, 1982. w13x S.F. Mughabghab, M. Divadeenam and N.E. Holden, Neutron Cross Sections ŽZ s 1]60., Vol. 1., Part A, Academic Press, New York, 1981. w14x R.S. Stein, Mol. Cryst. Liq. Cryst. 180A Ž1990. 119. w15x D.W. Schaefer, Science 243 Ž1989. 1023. w16x R. Vacher, T. Woignier, J. Phalippou, J. Pelous, E. Courtens, Rev. Phys. Appl., Coll. C4 24 Ž1989. C4]127. w17x M. Foret, J. Pelous, R. Vacher, J. Marignan, J. Non-cryst. Solids, 145 Ž1992. 133. w18x H.D. Bale, P.W. Schmidt, Phys. Rev. Lett. 53 Ž1984. 596. w19x B. Hammouda, S. Krueger, C.J. Glinka, J. Res. NIST 98 Ž1993. 31. w20x C.I. Merzbacher, J.G. Barker, K.E. Swider and D.R. Rolison, J. Non-cryst. Solids, 224 Ž1998. 92. w21x G. Dagan, M. Tomkiewicz, J. Phys. Chem. 97 Ž1993. 12651.