Estimation of mass thickness response of embedded aggregated silica nanospheres from high angle annular dark-field scanning transmission electron micrographs

Authors


Summary

In this study, we investigate the functional behaviour of the intensity in high-angle annular dark field scanning transmission electron micrograph images. The model material is a silica particle (20 nm) gel at 5 wt%. By assuming that the intensity response is monotonically increasing with increasing mass thickness of silica, an estimate of the functional form is calculated using a maximum likelihood approach. We conclude that a linear functional form of the intensity provides a fair estimate but that a power function is significantly better for estimating the amount of silica in the z-direction.

The work adds to the development of quantifying material properties from electron micrographs, especially in the field of tomography methods and three-dimensional quantitative structural characterization from a scanning transmission electron micrograph. It also provides means for direct three-dimensional quantitative structural characterization from a scanning transmission electron micrograph.

Lay Description

In this study we show that for certain systems it is possible to study the mass-thickness of micrographs retrieved with High Angle Annular Dark-Field Scanning Transmission Electron Micrographs without much knowledge about the microscopy setup. This can be done by an analysis using a log-likelihood model. We demonstrate our approach using theoretical as well as experimentally obtained aggregates of nano silica spheres. The work adds to the development of quantifying material properties from electron micrographs as well as estimating three-dimensional properties from two-dimensional micrographs.

Introduction

Material characterization using transmission electron microscopy is a very active field. In particular, tomography methods have shown to be powerful (Midgley & Dunin-Borkowski, 2009) for direct access to the three-dimensional structure and subsequent morphological analysis (Gommes et al., 2010). Of particular importance has been the development of the scanning transmission electron microscope (STEM) high angle annular dark field (HAADF), where the image intensity has an approximately monotonic relationship to the mass-thickness of the specimen (Hawkes, 2005) provided that the material studied is amorphous and that the collected electrons are incoherently scattered, which is known as the projection requirement (Midgley & Weyland, 2003). The development of electron tomography methods has stimulated an increasing number of studies of controlled fabrication (Chiappini et al., 2010), as well as characterization and reconstruction of materials in three dimensions (see e.g. Biermans et al., 2010; Saghi et al., 2011; or the study by Xin et al., 2010). Material reconstruction by electron microscopy micrographs has its roots in the 1980s, where the main challenge was to prepare sufficiently thin samples in order to approximate the slice to a representative two-dimensional cut through the material (see e.g. Weitz & Oliviera, 1984). It is worth emphasizing the possibilities of material characterization by direct thickness contrast imaging. However, to access this possibility, a functional form of the intensity response is needed first. In this study, we show that when the projection requirement holds, it is possible to directly estimate the intensity versus mass-thickness function from the micrographs using maximum likelihood. We demonstrate this approach using aggregates of nanometre sized silica particles where they have aggregated to form a (very stable) particle gel.

Materials and methods

The silica sol (BINDZIL 40/130) consisting of a 40 wt% aqueous dispersion of monodisperse silica spheres of diameter around 20 nm was kindly provided by EKA Akzo Nobel, Sweden. The sol pH was originally in the range of 9.10–9.20 and was adjusted to 7.8 by ion exchange (Dowex Marathon C, Sigma-Aldrich, Stockholm, Sweden) followed by suction filtration. The filtered sol, sodium chloride solution and deionized water were mixed, vortexed and left to gel for 14 days. Gel cubes of 1 × 1 × 1 mm were embedded in LR White resin (TAAB laboratories, Equipment Ltd., Berkshire, UK). The cubes were taken from the inner volume of the gel sample, discarding any surface areas. Prior to embedding, dehydration of the sample was performed in a graded ethanol series up to 99.5%. Ultra thin sections of approximately 90 nm (estimated from the colour of the reflectance of the section) were sliced using an ultramicrotome (Powertome XL, RMC products, Boeckeler Instruments Inc, Tucson, Arizona). The sections were placed on 200 mesh carbon support film Cu-grids and imaged in HAADF STEM mode with a Tecnai G2 (FEI Company, Eindhoven, the Netherlands) using an accelerating voltage of 200 kV and with a camera length of 300 mm giving a HAADF detector inner radius of 22 mrad.

Statistical analysis

Let us describe the sample by coordinates inline image so that inline image describes penetration axis and zmax indicates the thickness of the slice (in this study set to 90 nm). The mass thickness of the silica at inline image can be written as a fraction of the total sample thickness

math image(1)

where inline image denotes the silica density (depending on whether there is a silica particle at the point inline image or not). Using this, we can write an estimate of the intensity in the direction normal to the plane of the sample (i.e. the z-direction) as

display math(2)

where inline image and inline image denote the intensity response with respect to the mass thickness of silica and the embedding polymer, respectively. The goal of this study is to estimate the functional form of the intensity response inline image with respect to α and our ansatz is inline image where b, c and β are found by maximizing the log-likelihood function.

When the projection requirement holds, the intensity increases monotonically with increasing mass thickness, and the expression for the intensity can be expanded in powers of α. In particular, where the response from the embedding polymer is weak, an expansion of the silica thickness is sufficient for good estimates of the response. Taking into account a base level intensity b and a function describing the random noise inline image a simple model describing the observed intensity is

display math(3)

where c is a constant and g is below specified as a power function. We assume that the random noise inline image is well described by a normal distribution inline image and that noise from different pixels inline image are independent. Although this approach neglects diffraction effects (see e.g. Midgley & Weyland, 2003), we have found that under adequate conditions this approximates the intensity response in the system studied here well. By rewriting the fraction of silica the in z-direction α, as a function of particle centre positions inline image and defining a combined parameter-state vector inline image, the log-likelihood function inline image for the image data becomes

math image(4)

where inline image denotes the original micrograph intensity to be estimated, M is the set of pixels and |M| is the number of elements in the set M. Equation (4) can be maximized using standard methods (e.g. simulated annealing). To summarize, the log-likelihood function is maximized by varying the positions of a set of artificial particles (with the particle centres inline image) as well as varying the parameters (inline image). Such a configuration yields an artificial micrograph that is compared with the experimentally obtained image according to Eq. (4).

Results and discussion

The model and the approach outlined earlier was first tested using six generated micrographs where (true) parameters inline image and c were given random values. Furthermore, N spherical particles were given coordinates using a spatial Poisson process, with intensity varying from 10 to 30 (chosen from analysis of real micrographs to generate aggregates of similar density). One such micrograph I is shown in Figure 1 for the values inline image and inline image. The intensity was set to 30 and gave in this case inline image. The obtained micrograph I was then tested using an algorithm for maximizing Eq. (4) (i.e. the parameters and particle positions were estimated by maximizing the log-likelihood). The configuration of the estimated parameters is shown by the micrograph in Figure 2 (left-hand side). The absolute difference between the generated micrograph (Fig. 1) and the micrograph obtained by the maximization of Eq. (4) is shown in Figure 2 (right-hand side). The true parameters were successfully estimated in all the generated images. A similar analysis was then made for experimentally obtained micrographs.

Figure 1.

A generated micrograph using the model in Eq. (3) for inline image and inline image. A spatial Poisson process was used to assign inline image particle positions.

Figure 2.

Left image: A re-generated micrograph obtained by maximizing the log-likelihood function in Eq. (4) using the micrograph in Figure 1. The parameters were estimated to be inline image and inline image, which are to be compared with the true values used in Figure 1. Right image: A comparison of the particle positions by forming a residual image inline image of the original micrograph I0 (Fig. 1) and the re-generated one I shows a good estimate of the original micrograph.

A representative micrograph is presented in the top left corner of Figure 3 (top left-hand image) taken of 5.5 wt% nano silica prepared as described earlier. By directly measuring the intensity in the micrograph, it was noted that four rather separate values were obtained. These were attributed to the noise alone, or to one, two or three silica particles projected together, which was used as a starting value in the maximization of Eq. (4), where also the exponent β was initially set to one. By this maximization an estimate of the power function intensity response inline image was obtained with inline image, inline image and the constant inline image. This function is shown in Figure 4 (blue line) where the error bars show the estimated standard deviation of the noise σ. Also shown (black line) is an alternative model, where the intensity response was estimated using a linear function (i.e. β was kept at 1 and c was estimated). The two models with a power function response and linear response are nested and can thus be tested from the log-likelihood fits. The analysis in this paper is based on one single image, but to emulate repetitions we divided the image along three vertical and three horizontal lines to obtain 16 identical smaller rectangular images. To test if the power function response was better than the linear response we estimated β in each of these smaller images and computed the p-value in a test for the exponent being 1 or larger. Using a normal approximation and a corresponding t-test, we obtained the p-value, inline image showing with overwhelming significance that the power function is better than the linear response, which should also be clear from Figure 4.

Figure 3.

Top left side image: Micrograph of a 90 nm slice of 5 wt% aggregated nano silica obtained with HAADF-STEM. Top right side image: re-generated micrograph by maximizing the log-likelihood function (Eq. (4)) using a power function as described. Bottom left side image: Residual image inline image, of the original micrograph I0 and the re-generated one I. Bottom right side image: blow-up of the residual image showing the top left cluster.

Figure 4.

The intensity response I versus the amount of silica alpha as estimated by maximizing the log-likelihood function (Eq. (4)). The micrographs are STEM projections of 90 nm thick microtomed sample sections, where the individual silica spheres are 20 nm in diameter (See Fig. 3). One sphere corresponds to alpha = 0.22. The blue line shows an estimate using the inline image model, and the black line shows a linear model (inline image and c is estimated). The error bars show the estimated standard deviation of the background noise.

The top right micrograph in Figure 3 shows the resulting re-generated micrograph. In the bottom left part of Figure 3, the absolute difference between the experimentally obtained and the re-generated micrographs is shown. In Figure 5, a histogram of the intensity of the micrograph (blue line), the re-generated image (black line) and the estimated noise (red) are shown. The background pixel intensity is added (no negative pixel values), which is why the estimated noise is shifted from origo. The re-generated micrograph (blue line) underestimates the intensity significantly in the range between 0.1 and 0.2. From investigating the estimated intensity response (Fig. 5), it is hypothesized that this range corresponds to the edges of one particle. In Figure 3, the bottom right image shows a blow-up of the top left corner of the residual image, and indeed the errors seem to be located at the edges of the particles, which may partly be due to variation in particle size. Figure 5 also shows that the assumption of normal observation errors in Eq. (3) is reasonably adequate.

Figure 5.

Histogram of the pixel intensities for the micrograph (blue) and the generated micrograph (black). The estimated background noise is also shown (red). Note that the micrograph is more blurry at the edges of the silica spheres. This effect is not taken into account in the model, which can explain why the generated data show an underestimate just between inline image and 0.2.

Conclusions

We have shown that it is possible to retrieve the functional form of the intensity response dependence of mass thickness by direct analysis of the micrographs. For this procedure to work, a well-defined material such as the mono-disperse and mono-phase particles or, as in this case, aggregates of particles must be used.

By knowing the mass thickness-intensity function, a three-dimensional estimate of the studied sample can be made from the micrograph. The only assumption needed regarding the material structure is that all particles have aggregated (and that specimen surface effects from the thin film TEM sample preparation resulting in partial aggregates are avoided).

Acknowledgements

The authors would like to thank Stefan Gustafsson (Chalmers University of Technology) for fruitful discussions and valuable comments. The work was funded by the Swedish Science Council (VR project no. 2008–3895) and the Vinnova financed VINN Excellence Center SuMo Biomaterials.

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