**NanoMet: You Can’t Measure Changes This Subtle By Hand**

Silicon dioxide (SiO_{2}) is an interesting test case for NanoMet because it forms nearly perfect spherical particles. They all look very similar by eye but have subtle differences that are easy to discern with Nano Met’s Particle Diameter Module. In this application note, silica nanoparticles were made through the hydrolysis of TEOS (tetraethyl orthosilicate) in ethanol using ammonium hydroxide to control the particle size. This is a common and widely accepted method for SiO_{2} nanoparticle synthesis.

Nanoparticles have a low Reynolds number when suspended in fluid and in this Stokes flow regime particles have a terminal velocity that is proportional to their size as they settle in gravity. This allows particles to be size-separated according to sedimentation time, a principle used to separate clay fractions in mineralogical samples and to measure macromolecular sizes using an ultracentrifuge. In this study SiO_{2} nanoparticles were subject to sedimentation. The synthesized material was sonicated and vortexed, and 10 mL of 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, dried, and then coated with 4 nm of iridium for scanning electron microscopy (SEM) imaging.

Spherical particles tend to hexagonally close-pack, but in the case of this sample the size distribution is broad enough that the particles do not closely pack in well-defined layer. To accurately visualize the particles for this test case, a sample preparation was developed to achieve a single-layer dispersion of material for imaging. To best leverage NanoMet’s ability to provide interactive control of image segmentation, the sample was imaged at 10 kV using backscattered electrons (BSED) to minimize contrast gradients across the particle surface due to the edge effects commonly seen in secondary electron imaging. Four images were taken at the same magnification (30 kX) and the same SEM contrast settings for the unsettled material and material settled after 60 minutes.

The images above show the unsettled (left) and material allowed to settle for 60 minutes (right). ** There is no obvious difference to the naked eye and a set of ten or twenty randomly selected measurements fails to provide any conclusive difference in the average size of the particles. **The four images for each sample were uploaded to the NanoMet cloud platform that recognizes and the scale bar in each image allowing for simple and fast

*automatic pixel size calibration*. The images were then subject to particle size analysis using the NanoMet Particle Diameter Module. Each image yielded 300-400 measurements in around 2 seconds. The measurements from each image were saved as image metadata allowing the NanoMet

*batch reporting capability*to combine the statistics for all four images for more rigorous analysis using all 1500+ measurements from each sample. Since the same SEM brightness and contrast settings were used to image all four images, this analysis was a trivial task for NanoMet given its capability to apply the module parameters across batches of images.

For the unsettled sample, NanoMet measured 1862 diameters to give a mean diameter of 342.2 nm with a standard deviation of 26.4 nm. The third and fourth statistical moments, skew (-0.65) and kurtosis (0.99), show that the particle size distribution is not normal. The histogram to the left shows the unsettled particle size distribution as a densely populated and well-binned graph, which is a fully automated deliverable from the* report generation capabilities* of NanoMet. The distribution is clearly skewed towards smaller particle sizes, which is expected as they will settle more slowly under gravity. This was impossible to determine looking at the image or through a handful of manual measurements. Both the NanoMet cloud platform and FullScaleNANO’s Histogram On Demand (HOD) service provide these types of histograms of particle sizes.

The metrology of nanoparticles is important not only to characterize the materials themselves, but also the effects of storage and handling. Nanoparticles are unique in that they have a very large ratio of surface area to volume compared to bulk materials, so ambient conditions can modify the nature of nanoparticles through decomposition processes such as oxidation. In the case of hydroscopic materials, nanoparticles aggregate into larger clusters when exposed to air. The metrology of nanoparticles is even critical to understanding their transport in gas or liquid phase as they are subject heavily to size-dependent separation.

The particle size distribution of the test case sample that settled for 60 minutes is shown to the right. Again, this histogram is shown as generated by NanoMet’s automatic report generation capability. Histogram On Demand users would receive similar histograms as part of their on-demand service. For the 60 minute settled sample 1679 diameters were measured with a mean diameter of 339.2 nm and 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 shows the particle diameter distribution is more “normal-like” and definitively less broad. Here we can see the effect of a short period of settling on a nanoparticle size distributions within a dynamically changing suspension.

** NanoMet’s capability of measuring thousands of objects in seconds allows the full power of statistics to be used for analysis and quality control. **The large number of measurements, N, allows the precise determination of confidence intervals through the following equation

where is the estimated mean diameter from statistical sampling, the corresponding standard deviation, the confidence coefficient for the confidence level, and the actual mean. These z-scores or confidence coefficients are well tabulated, and for a 96% confidence level, z_{c} = 2.05. In the case of the unsettled sample, this large sample size gives a diameter of 342.22 ± 1.26 nm, and for the sample settled for 60 minutes, a diameter of 339.17 ± 1.48 nm. *These intervals do not overlap, so with 96% confidence we know that these two samples, despite being of the same synthesized material,**are statistically different with respect to size*. ** **

With the large number of measurements and the quality of the histograms produced by both FullScaleNANO’s NanoMet software and Histogram On Demand (HOD) service, *advanced nonparametric tests *can be performed 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. That is obvious by eye looking at the histograms above, but it can also be determined analytically (with confidence intervals) using the large volume of data that NanoMet produces with the click of a button.

The K-S test was quite easily performed on the two samples in this app note as NanoMet outputs the raw data it measures in a universal spreadsheet format. The particle distributions were converted to cumulative probability distributions, F(d), by integrating and normalizing the particle diameter histograms shown above. The cumulative probability distributions are very similar except in the range of 270-330 nm, and it is this deviation that the K-S test exploits. The formalism of the K-S test can be expressed as follows

where: c(a) is a parameter related to the K-S test that depends 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(a) 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. *From this K-S test we know with 99.999% confidence that these two samplings of SiO _{2} nanoparticles are in fact, represented by different statistical distributions.*

These two samples differed in average diameter by only 3 nm but this was quickly and conclusively determined by NanoMet in seconds. Conducting the two different statistical tests* was possible only because of the large number of diameter measurements provided by NanoMet and FullScaleNANO’s Histogram On Demand (HOD) service*. The large number of samples significantly reduced the confidence intervals to show that the average particle diameters were meaningfully different within 96% confidence using z-scores. And the ability to generate dense histograms with 1500+ measurements for each sample allowed a nonparametric K-S test to show within 99.999% confidence that these samples had different distributions of diameters. This was a clean, fast and easy task with NanoMet but the same just cannot be said for any statistical results coming from sketchy selective measurements done by hand.