**Silicon dioxide (SiO**_{2}) forms near-perfect spherical particles, making it an interesting test case for NanoMet. These particles look very similar when viewed by the naked eye, but they can be easily differentiated by using Nano Met’s Particle Diameter Module.

In this article, silica nanoparticles were synthesized by the hydrolysis of TEOS (tetraethyl orthosilicate) in ethanol using ammonium hydroxide to control the particle size. This method is commonly used to synthesize SiO_{2} nanoparticles.

Nanoparticles possess a low Reynolds number when suspended in fluid and in this Stokes flow regime, the terminal velocity of particles is proportional to their size as they settle in gravity. As a result, particles can be size-separated based on sedimentation time, a principle employed to separate clay fractions in mineralogical samples and to determine macromolecular sizes using an ultracentrifuge.

## Experimental Procedure

In this study, SiO_{2} nanoparticles were subject to sedimentation. After sonicating and vortexing the synthesized material, 10 µL of the material was drawn from the same depth of the vial immediately after agitation and one hour after the material was allowed to settle. The sample material was pipetted onto glass, allowed to dry, and then coated with 4 nm of iridium for scanning electron microscopy (SEM) imaging.

Spherical particles have the tendency to hexagonally close-pack, but the broad size distribution of this sample prevents the particles from closely packing in well-defined layer. For this test case, a sample preparation was developed to achieve a single-layer dispersion of material to accurately visualize the particles.

To fully exploit NanoMet’s ability to provide an interactive control of image segmentation, the sample was imaged at 10 kV using backscattered electrons (BSED) to minimize contrast gradients across the particle surface caused by the edge effects prevalent in secondary electron imaging.

Four images were captured at the same magnification (30 kX) and the same SEM contrast settings for the unsettled material and material settled after 60 minutes.

## Experimental Results

The images above reveal the unsettled (top) and material allowed to settle for 60 minutes (bottom). They all look very similar to the naked eye and a set of 10 or 20 randomly selected measurements fail to provide any conclusive difference in the average particle size.

The four images for each sample were uploaded to the NanoMet cloud platform that recognizes the scale bar in each image, enabling simple and fast automatic pixel size calibration. The NanoMet Particle Diameter Module was used to subject the images to particle size analysis.

Each image yielded 300 - 400 measurements within 2 seconds. The measurements from each image were recorded as image metadata, enabling the NanoMet batch reporting capability to combine the statistics for all four images to perform more rigorous analysis using all 1500+ measurements from each sample.

As all four images were imaged using the SEM brightness and contrast settings, this analysis was a trivial task for NanoMet, considering its capability to apply the module parameters across batches of images.

For the unsettled sample, NanoMet measured 1862 diameters, and the mean diameter was found to be 342.2 nm with a standard deviation of 26.4 nm. The third and fourth statistical moments, skew (-0.65) and kurtosis (0.99), reveal that the particle size distribution is not normal.

The histogram above reveals the unsettled particle size distribution as a densely populated and well-binned graph, which is a completely automated deliverable from the report generation capabilities of NanoMet. The distribution is skewed towards smaller particle sizes as expected because they will settle more slowly under gravity.

This was not possible to determine looking at the image or through a handful of manual measurements. These types of histograms of particle sizes are provided by both the NanoMet cloud platform and FullScaleNANO’s Histogram On Demand (HOD) service.

The metrology of nanoparticles is important for characterizing the materials themselves, and also for the effects of storage and for handling. As nanoparticles have a larger ratio of surface area to volume than bulk materials, their nature can be modified by ambient conditions through decomposition processes such as oxidation.

In the case of hydroscopic materials, nanoparticles tend to aggregate into larger clusters upon exposure to air. The metrology of nanoparticles is even critical to gain insight into their transport in gas or liquid phase as they are subject heavily to size-dependent separation.

The above histogram shows the particle size distribution of the test case sample settled after 60 minutes. Again this image is shown as generated by NanoMet’s automatic report generation capability. As part of their on-demand service, Histogram On Demand users would obtain similar histograms.

For the 60 minute settled sample, NanoMet measured 1679 diameters and the mean diameter was found to be 339.2 nm with a standard deviation of 29.6 nm. The distribution is still skewed towards smaller diameters as expected, with a third moment (skew) of -0.76.

The fourth moment (kurtosis) of a normal distribution is 3, and the shift of the kurtosis from 0.99 in the unsettled sample to 1.39 in the 60 minute settled sample reveals that the particle diameter distribution is more “normal-like” and clearly less broad. Here, the effect of a short period of settling on a nanoparticle size distributions can be clearly seen within a dynamically changing suspension.

NanoMet’s capability of measuring thousands of objects within seconds enables leveraging the full power of statistics for analysis and quality control.

The large number of measurements (N) allows the precise determination of confidence intervals using the following equation:

Where, *x* = Estimated mean diameter from statistical sampling, σ = Corresponding standard deviation, z_{c} = Confidence coefficient for the confidence level, and µ = Actual mean.

These z-scores or confidence coefficients are well tabulated, and for a 96% confidence level, z_{c} = 2.05. For the unsettled sample, this large sample size provides a diameter of 342.22 ± 1.26 nm, and for the sample settled after 60 minutes, a diameter of 339.17 ± 1.48 nm.

As these intervals do not overlap, these two samples, with 96% confidence, are statistically different with respect to size though they were from the same synthesized material.

With the large number of measurements and the quality of the histograms generated by both FullScaleNANO’s NanoMet software and Histogram On Demand (HOD) service, it is possible to perform advanced nonparametric tests on particle sized distributions.

One such test is the Komolgorov-Smirnov or K-S test, which can be used to determine if two samples came from the same statistical distribution. This can be seen clearly by looking at the histograms above. However, it can also be analytically determined (with confidence intervals) using the huge volume of data produced by NanoMet with the click of a button.

As NanoMet outputs the raw data measured by it in a universal spreadsheet format, it was easy to perform the K-S test on the two samples. The particle diameter histograms shown above were integrated and normalized to convert the particle distributions to cumulative probability distributions, F(d). The cumulative probability distributions are very similar except in the range of 270 - 330 nm, and this deviation is exploited by the K-S test. The formalism of the K-S test can be expressed as follows:

Where, c(α) is a parameter related to the K-S test depending upon the confidence level; N_{0 min} and N_{60 min} are the number of measurements for the unsettled and 60 minute settled sample, respectively; and the right hand side of the equation represents the maximum separation between the two cumulative probability distributions.

For a 99.999% confidence interval, c(α) is 1.95 making the left side of the inequality 0.0022. The maximum deviation between the cumulative probability distributions occurs around 320 nm, and is 0.046. The K-S test results reveal that with 99.999% confidence, the two samplings of SiO_{2} nanoparticles are indeed represented by different statistical distributions.

## Conclusion

The average diameter of these two samples differed by only 3 nm, but NanoMet was able to quickly and conclusively determine within seconds. Performing the two different statistical tests was only possible due to the capability of NanoMet and FullScaleNANO’s Histogram On Demand (HOD) service to provide a large number of diameter measurements.

Using the large number of samples considerably reduced the confidence intervals to demonstrate that the average particle diameters were meaningfully different within 96% confidence using z-scores. Also, the ability to produce dense histograms with 1500+ measurements for each sample allowed a nonparametric K-S test to show that within 99.999% confidence, these samples had different distributions of diameters.

This task was clean, quick and easy with NanoMet, but the same cannot be said for any statistical results coming from sketchy selective measurements carried out by hand.

This information has been sourced, reviewed and adapted from materials provided by FullScaleNANO.

For more information on this source, please visit FullScaleNANO.