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In-situ SAXS studies of the morphological changes of an alumina-zirconia-silicate ceramic during its formation

D Le Messurier, R Winter, CM Martin; J Appl Cryst, 39 (2006) 589
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Abstract

Small-Angle X-ray Scattering (SAXS) is used at two energies, one either side of the zirconium K-edge, to probe the in-situ formation of an alumina-zirconia-silicate ceramic. The use of energies either side of the edge allows the decomposition of information regarding the scattering from the zirconia particles from that of the glass matrix. Porod slope data shows how the nanoparticles progress from being relatively isolated particles to becoming agglomerates as the pore network in the glass collapses. The shape of the agglomerates resembles the pore network of the glass at low temperature. The Guinier radii of the particles show the growth of the agglomerates past the Littleton softening point, whilst still resolving the primary particles.

Introduction

Small-Angle X-ray Scattering (SAXS) is used at multiple energies to probe the in-situ formation of an alumina-zirconia-silicate ceramic. By choosing energies below the zirconium K-edge (17.98keV) and above the edge (18.05keV) we are able to distinguish between the zirconia particles and the surrounding glass matrix. The energies have been chosen such that the zirconium signal is attenuated in the scattering pattern obtained above the absorption edge. By having the zirconium signal reduced in one of the scattering patterns we are able to infer the contribution from the zirconia particles and thus distinguish the scattering from the glass matrix from that of the zirconia particles.
Our sample consists of a collection of nanoparticles and a powdered glass matrix. These components have been pressed together to form a pellet in order to conduct the SAXS experiment. The sample could therefore be considered as a collection of particles being situated in the pores of some porous medium, as the nanoparticles fill the gaps where several macroscopic glass grains touch. In this case the scattering function,S(q), can be described in terms of one of two fractal models, surface fractal and mass fractal. For a surface fractal of fractal dimension, Ds, the scattering function can be expressed as:
$S(q)=\frac{\pi(\Delta\rho)^2S_2l_2^{D_s-2}\Gamma(5-D_s)\sin[\pi(3-D_s)/2]}{(3-D_s)q^{6-D_s}}$
with S₂ being the smooth surface area, measured at a length scale l₂ where the surface fractal behaviour cuts off, and Gamma is the gamma function. The scattering function for a mass fractal of mass fractal dimension, D can be expressed as:
$S(q)=NF^2(q)(\Delta\rho)^2\bigg[1+\frac{1}{(qr)^D}\frac{D\Gamma(D-1)}{[1+\frac{1}{q^2\xi^2}]^{(D-1)/2}}\times\sin[(D-1)\tan^{-1}(q\xi)]\bigg]$
where xi is the fractal correlation length which represents a characteristic distance above which the mass distribution in the sample is no longer described by a fractal law. N is the total number of particles, F(q) is the average form factor, Delta rho is the scattering contrast defined as
$(\Delta\rho)^2\propto f^*f=f_0^2+2f_0f'+(f')^2+(f'')^2$
In this formalism of the scattering contrast, f₀² is the normal SAXS signal, 2f₀f' is the cross term, and (f')²+(f")² is the resonant term.
Here we will simplify the analysis of the SAXS patterns obtained by using a simple power law of the form:
$I(q)=I_0q^{-\alpha}$
where I₀ and alpha are constants. The values of alpha can be determined from the slope of the Porod regime of a log(I(q)) vs. log(q) plot. From these values, the mass (D) and surface (D_s) fractal dimension can be calculated. For a mass fractal, alpha = D, so 1 < alpha < 3 since 1 < D < 3, whereas for a surface fractal, alpha = 6-D_s so 3 < alpha < 4 since 2 < D_s < 3.
One other important parameter which can be obtained from the SAXS scattering patterns is the Guinier radius, R_g. The Guinier radius is linked to the radius of gyration of the constituents which contribute to the scattering contrast by a shape dependent prefactor, as long as the particle sizes are in the range 1/q_max to 1/q_min. This is seen experimentally as a "hump" in the scattering pattern plotted in log-log format, which can be reproduced by using an exponential function of the form:
$I(q)=A\exp\Big{\{}\frac{R_g^2q^2}{3}\Big{\}}+C$
where A and C are constants, and R_g is the Guinier radius.

Experimental

SAXS patterns were taken at energies of 17.98 and 18.05keV alternately during the same heating cycle, with the temperature increasing from 350 to 725^oC at intervals of 25K.
The choice of two energies on opposite sides of the edge instead of the standard anomalous SAXS experiment, where several energies just below the edge and one far removed from it are used, is due to the difficulty in determining the accurate position of the edge. There is a chemical shift of the edge position of approximately 8eV between our sample and Zr foil, which changes by several eV as the experiment progresses. As an accurate knowledge of the position of the edge is critical in ASAXS to work out the exact contrast as a function of energy, we settled for the largest obtainable contrast between above and below the edge, which is smaller but not subject to error due to the edge shift. This contrast can easily be seen in the scattering patterns obtained from above and below the edge.

Fig.: Differential edge scans before and after heating.

Scanning the energy accurately reveals the location of the edge. There is a chemical shift between the un-annealed (red) and the annealed (blue) sample.

Fig.: Scattering patterns above and below the edge.

The scattering patterns obtained at energies of 17.98keV (red) and 18.05keV (blue) and a temperature of 450^oC.

Since our sample is initially a pressed pellet, its density will change during heating due to the removal of pores and, to a lesser extent, thermal expansion. An indication of the amount of densification can be obtained by an application of Beer's law. An indication of the densification of the sample can be obtained by plotting the product of density and thickness against temperature and measuring the initial and final thickness' of the sample to see by how much the sample thickness has altered. From 450^oC to 500^oC (just below the Littleton softening point) there is a plateau region in the general increase of density*thickness, indicating no significant change in physical sample dimensions. At 600^oC the maximum value of density*thickness has been achieved, and at higher temperatures the density*thickness product starts to decrease again.

Fig.: Product of sample density and thickness as function of temperature.

The variation of sample density times thickness as the temperature increases during sample heating.

Results

The scattering patterns obtained need to be subjected to some corrections before analysis of the Porod slopes and Guinier radii can be undertaken. It is necessary to correct the background scattering patterns for the sample transmission at each temperature. Although the background was taken at room temperature, the sample transmission changes during heating due to the densification of the sample. Therefore, a transmission-corrected background signal is subtracted from the high-temperature runs at each energy. There is some fluorescence in the scattering patterns obtained at 18.05keV arising from the 10.06 atom\% of zirconium within the sample. This fluorescence has no angular dependence and has not been removed from the scattering patterns but tthe regions used for analysis have not extended into the high q limit near the edge of the detector, where fluorescence is dominant.

Fig.: Typical scattering pattern and Guinier and Porod fit components.

The scattering pattern obtained at 450^oC (blue), fit components (red; A-Guinier and B-Porod), and unified model fit (green).

Guinier plot showing the 375^oC, 17.98keV data set and the Guinier fit component.

The slope analysis of the Porod regime was achieved by weighted least squares regression of the Porod equation (curve B). As a consequence of the high energies used to create the scattering contrast we have a very compressed q scale which does unfortunately give a very short Porod region. The region used for fitting is limited to the linear section and does not extend into the detector range where there would be a high fluorescence contribution.

The Guinier radii were obtained by fitting a unified Guinier exponential/power-law model to the scattering pattern. The unified model, in essence, is a superposition of the Porod and Guinier equations and is an approximate form that describes the complex morphology of the sample over a range of structural levels. A structural level in scattering is described by Guinier's equation (curve A) and a structurally limited power law (Porod eq.), which on a log-log plot is reflected by a knee and a linear region. Displaying the fit over the knee in the log-log plot as a traditional Guinier plot shows the quality of the fit used.

Discussion

At higher temperatures, the scattering patterns exhibit a clear maximum in the log-log plot of I(q), indicating particle-particle interactions are present. This greatly complicates the analysis as most formalisms are not strictly valid except in the dilute solution limit. With these points in mind, the values obtained from the Guinier analysis are subject to systematic errors. However, in-situ experiments are concerned with trends on variation of a parameter, such as temperature, rather than absolute values, and clear trends emerge from this type of analysis.	The 17.98keV scattering patterns at 350^oC (red), 450^oC (green), 550^oC (blue), 650^oC (pink), and 725^oC (light blue).
Guinier radii measured at 17.98kev (stars) and 18.05keV (triangles). A second Guinier radius emerges above 600^oC.	Porod slope exponents obtained at 17.98keV (stars) and 18.05keV (triangles).
At low temperatures the values of R_g obtained at the two energies are very close to each other. Then, at a temperature between 525^oC and 550^oC, there is a splitting of the line. This splitting indicates the appearance of a second Guinier region, i.e. the formation of a second hump in the scattering pattern. The point at which the values of R_g split is attributed to the dilatometric softening point of the glass, which is defined as the temperature at which the glass deforms under its own weight. The generally quoted associated temperature is known as the Littleton softening point, i.e. the temperature at which the viscosity is 10^7.6dPas. For our glass composition the theoretical value of the Littleton softening point is 503^oC. The splitting of R_g is attributed to the growth of agglomerates in the sample. Past the Littleton softening point the glass starts to soften, and the gaps between the glass grains start to change shape. Since the nanocomposite sits in these gaps the nanoparticles start to move and form agglomerates. This also explains the constant R_g branch, as this would represent individual primary particles.	The temperature dependence of the slope of the scattering pattern is more complex and requires thought about the change in morphology of the sample as heating progresses. In the Debye model the treatment depends on the assumption of atomically smooth boundaries between the inhomogeneities and the host material. Also, in the limit of large scattering vectors, in the size range xi < q < r₀, where r₀ is the typical inter-atomic distance, the Debye model predicts the characteristic q^-4 Porod power-law behaviour. Again, this requires that the interfaces are sensed as smooth on an atomic scale. In practice we observe that none of our data obey this law, though all have a power-law, q^-n dependence, where n(which we call the Porod exponent) is non-integral and less than 4. Non-integral power-law scattering in the Porod region may also be exhibited by systems in which the size distribution of the scattering inhomogeneities is itself a power law, with scattering proportional to q^-n arising from a particle (or pore) size distribution proportional to r². However, particles grown by condensation processes such as sol-gel and vapour deposition techniques are usually subject to a log-normal size distribution because the reaction cross-section depends on the surface to volume ratio of the particles. The most simplistic model which fits the data and our knowledge of the initial microstructure of the sample is that of a combination of surface and mass fractals. It is with this model in mind that we interpret our data.

The dominating contrast in the 18.05keV data below 500^oC is between glass and air, since there will be few nanoparticles in contact with the glass compared with the contact area between glass and pore (air). The geometry of this glass-pore network can be described by using a mass fractal model. The 18.05keV datasets can be regarded as containing a minimal amount of information about the zirconia nanoparticles since the scattering arising from the glass (and the alumina nanoparticles) is largely unaffected by the change in energy. As the temperature increases, the rough surfaces of the glass grains start to smoothen as the energy required for deformation on the atomic length scale is surpassed. Above 500^oC the glass starts to soften even on a larger length scale, and the grain cores start to deform under their own weight, thus the pores reduce in size pushing the nanoparticles together into denser packed agglomerates. This causes the dominant contrast at 18.05keV to change and become a contrast between glass-nanoparticles-air. As the temperature rises above 600^oC the contrast changes to become solely between the glass and the nanoparticles.
The 17.98keV data is different since it contains full information regarding the zirconia particles. Below 500^oC the dominating contrast is between the zirconia nanoparticles and air, since the nanoparticles sit in the pores of the glass powder. Above 500^oC there is a transition from being dominated by a contrast between nanoparticles and air to a contrast between nanoparticles and glass. This is due to the onset of a larger scale deformation of the glass around the nanoparticles thus reducing the pore size and forcing the nanoparticles to compact. Above 600^oC the glass has deformed significantly such that the nanoparticles have been fully compacted into agglomerates with no pores. The fractal nature of the agglomerates at high temperature is due to the nanoparticle agglomerates taking a shape similar to the initial pore structure. This situation differs from the original fractal nature of the pores since the fractal correlation length is different due to the fractal structure being built from small primary particles.

The first box in the Figure (350^oC) shows the initial morphology of the sample, where the nanoparticles are sitting loosely in the pores and the glass grains have a rough surface. As temperature increases (450^oC box) the glass grains start to become less rough on the surface, as the energy necessary for movement on an atomic scale is surpassed. With further heating past the Littleton softening point (550^oC box) there is deformation on a much larger scale, causing the pores to collapse. This causes the nanoparticles to be pushed together to form agglomerates. At high temperatures (650^oC) the majority of the pores have been removed from the sample leaving the nanoparticles in agglomerates whose shape resembles the initial pore network.

Fig.: Schematic representation of the proposed model.

Schematic of the proposed model.

Conclusions

The formation of an alumina-zirconia-silicate ceramic has been investigated using in-situ SAXS at two different energies, one above and one below the zirconium K-edge. It has been shown that past the Littleton softening point of the glass, the nanoparticles start to arrange themselves to form agglomerates in the pores of the glass. By taking the Porod regime slope one can obtain information regarding the fractal nature of the components of the sample.
The 17.98keV data shows how the nanoparticles initially have a rough surface, and that as the temperature increases, they are forced together by the collapse of the pores to form agglomerates which resemble the shape of the initial pore network of the glass. The 18.05keV Porod slope data shows how the glass starts as a collection of jagged shaped grains, then as heating progresses the glass grains soften and congeal to form a continuous glass matrix. By looking at the Guinier radii at 17.98keV one can see the growth of agglomerates past the Littleton softening point, while the primary particles can still be resolved.

Acknowledgements

The authors would like to thank Chiu Tang, then station scientist at beamline 6.2 of the SRS in Daresbury, who is now at the Diamond Light Source. Armin Hoell of the Hahn-Meitner-Institut Berlin for his assistance during the data analysis. Also, Daniel Le Messurier would like to thank Pilkington Plc and EPSRC for a studentship from the Collaborative Awards in Science and Engineering scheme, and CCLRC for beamtime.

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