2.1.

Fundamental Principles and Limitations

The ideal microscopy for imaging and measuring nanoparticles can be imagined as one that could provide a three-dimensional (3-D) field of view so that the entire surface of each particle could be accurately mapped [Fig. 1(a)]. Such a microscopy technique could fully characterize the morphology of each particle and provide parameters for an ensemble of particles such as average surface area, volume, maximum and minimum dimensions, and diameter. Being a nonideal microscopy technique, AFM can only approach this imaginary technique.

AFM creates an image of a sample’s surface topography by mapping a surface of constant tip-sample interaction. To image a nanoparticle and measure its dimensional properties, it is necessary for the particle to be sitting on a flat substrate. The AFM only images the upwards facing surface of the particle as a topographical surface feature [Fig. 1(b)]. A further limitation is that an AFM tip has a finite radius of curvature at its apex which is comparable with the dimensions of a nanoparticle. Since the image generated by scanning the tip across the particle represents a convolution of the tip shape with the particle topography, this results in a broadening of the apparent lateral dimensions of the particle image [Fig. 1(c)]. Reducing tip diameter reduces this effect; for example, single-walled carbon nanotubes with sub-nanometer diameter have been used as AFM tips achieving lateral resolutions down to 2 nm.³

Fig. 1

(a) Imaginary ideal microscopy which maps the entire surface of the particle; (b) for AFM, the particle has to sit on a flat substrate, and only the upward facing surface is imaged; (c) the finite radius of the AFM tip results in a lateral broadening of the particle image. Therefore, assuming the particle is spherical, the diameter of the particle can most accurately be derived from its height.

2.2.

Deriving Particle Diameter from Height

Given these limitations, it is advisable to derive the particle diameter from the difference in height between the level of the substrate and the apex of the particle image [Fig. 1(c)], assuming that the particle is spherical.⁴ Nanoparticle reference materials are engineered for sphericity, so this assumption is not unreasonable. Furthermore, there are advantages in deriving the particle diameter from a height measurement: firstly, AFMs have a resolution in the vertical (out of plane) axis an order of magnitude better than their lateral resolution, and, secondly, a histogram method (see below) can be applied to readily obtain good statistics on the measurements derived from the AFM image.

The histogram method is effective for measuring height differences in AFM images and is one of the methods described and recommended in the VDI/VDE standard 2656, Determination of geometrical quantities by using Scanning Probe Microscopes—Calibration of measurement systems, for the calibration of step height transfer artefacts.⁵ The method treats the vertical axis value of each of pixel in the AFM image as an individual height measurement. Given the large number of pixels in an AFM image, mean height values of nominally flat surfaces, such as substrates, can be determined with good statistical robustness.

For an idealized AFM image [Fig. 2(a)], with scan range $a\times a$, of a single particle with a diameter $D$, sitting on a perfectly flat substrate, the resulting histogram is shown in Fig. 2(b). The pixel count, $n(h)$, of the substrate peak in the histogram is: $n(0)\propto [a^2-\pi {(D/2)}^2]$ and the distribution of the pixel heights from the particle is: $n(h)\propto \pi (Dh-h^2)$ between $h=D/2$ and $h=D$, where $h$ is the pixel height. The actual numbers of pixel counts plotted in the histogram will depend on the number of pixels in the image and the binning in the histogram. For an actual AFM image [Fig. 2(c)] of a 100-nm nominal diameter particle, the histogram can only be approximated by this model. The substrate will not be perfectly flat and will appear as a broadened peak; the lateral broadening of the particle image will result in a distribution of pixel heights which fills in the gap between the peak corresponding to the substrate pixels and the particle pixels; the particle may not be perfectly spherical (for example, a flattened particle will result in a steeper height distribution curve); and imaging noise will be superimposed across the whole pixel height distribution. These effects can be seen in the distribution of pixel heights in Fig. 2(d), where the histogram shows a maximum height slightly less than the nominal 100 nm diameter of the particle.

Fig. 2

(a) Ideal model of an AFM image of a particle on a flat substrate; (b) histogram showing the distribution of pixel heights of the particle imaged in (a), with the pixels of the substrate in blue and the pixels from the particle in orange; (c) rendered 3-D image of an actual AFM scan of a particle with a nominal diameter of 100 nm; and (d) histogram showing the distribution of pixel heights of the particle image in (c).

3.1.

Sample Preparation

Number density gradients of gold nanoparticles were created by first depositing a thin film of plasma polymer allylamine on 13-mm-diameter glass cover slips followed by time-controlled electrostatic binding the gold nanoparticles and the amine functionalized surface. Plasma polymerization was carried out in a custom-built reactor described elsewhere⁷ using a 13.56 MHz plasma generator. Deposition of allylamine was carried out at a flow rate of 10 standard cubic centimeters per minute, power of 10 W, and time of deposition of 4 min. These conditions result in nitrogen-rich films with a thickness of about 20 nm, as previously reported. Before deposition, all substrates were cleaned by oxygen plasma for 2 min using a power of 20 W.

The nominally 16-nm-diameter gold nanoparticles were synthesized by reduction of 50 ml of 0.01% ${\mathrm{HAuCl}}_4$ solution by adding 1 ml of a 1% sodium citrate solution at 100 °C. After cooling the resulting suspension to room temperature, the particles were surface-modified with 2-mercaptosuccinic acid.⁸

Over a period of 2 h, the functionalized amine substrates were gradually dipped into the undiluted suspension of 16 nm gold nanoparticles. The process was controlled by a Zaber T-LSR* linear motion drive operated with Zaber software.⁹ The particle number density gradient resulted from the length of time the substrate was exposed to the gold nanoparticle suspension as it was gradually immersed. The high-density end of the substrate received the maximum length exposure whilst the low-density end received the least.

3.2.

Imaging the Particle Number Density Gradient

Images were acquired with an Asylum Research MFP-3D AFM using intermittent contact mode. Initial image processing included an iterative flattening process involving a first-order line flatten, calculating a mask, followed by another first-order line flatten, and repeating the process until there was no perceptible change in the image. Image analysis was performed using the Scanning Probe Image Processor (SPIP™) software produced by Image Metrology A/S.¹⁰

The sequence of $1\mathrm{\mu m}\times 1\mathrm{\mu m}$ scan range images in Figs. 3(a)–3(e) illustrates how the particle number density increases across the sample. At the first position, (a), approximately 4 mm from the edge of the substrate, there are no particles and the image (with an amplified color scale) reveals the texture of the functionalized substrate. The histogram of this image [Fig. 4(a)] indicates that the height variation of the substrate is Gaussian with a standard distribution of $\sim 0.22\mathrm{nm}$. At the 5 mm point, (b), there are a few particles, and the density can be seen to increase in the successive images (c) to (f). Some particle agglomeration is visible in the images, however, this is limited to doublet and triplet particle clusters.

Fig. 3

Images acquired with $1\mathrm{\mu m}\times 1\mathrm{\mu m}$ scan range of regions across the gradient sample: (a) approximately 4 mm from the edge of the sample, there are no particles and only the substrate topography is visible (color scale: $\text{black}=-1.0\mathrm{nm}$ to $\text{yellow}=1.0\mathrm{nm}$), (b) at $\sim 5\mathrm{mm}$, there are $14\text{particles}/\text{clusters}$, (c) at $\sim 6\mathrm{mm}$, there are $41\text{particles}/\text{clusters}$, (d) at $\sim 8\mathrm{mm}$, there are $81\text{particles}/\text{clusters}$, (e) at $\sim 10\mathrm{mm}$, there are $137\text{particles}/\text{clusters}$, and (f) at $\sim 12\mathrm{mm}$, there are $177\text{particles}/\text{clusters}$ [color scale for images (b) to (f): $\text{red}=0\mathrm{nm}$ to $\text{white}=\sim 25\mathrm{nm}$].

Fig. 4

(a) Histogram of images Fig. 3(a) showing the spread of heights for the functionalized substrate with $\mathrm{SD}\approx 0.22\mathrm{nm}$ highlighted, (b) histogram of image Fig. 3(e), and (c) histogram of image Fig. 3(f). The red dotted line in (b) and (c) indicates the mean particle height values in Table 1 including clusters, the green dotted line indicates half the mean particle heights which should correspond to the $h=D/2$ point in Fig. 2(b).

3.3.

Image Analysis and Particle Height Measurement

Each of the images with particles, Figs. 3(b)–3(f), was analyzed using SPIP™’s particle analysis function to measure the heights of the particles. The particle analysis function uses a masking process to identify particles on the substrate and then calculates a wide range of particle size parameters for each particle, and measurement statistics for the set. Most of the parameters, however, rely on the accurate lateral measurement of the particles. The particle analysis function was used here to identify the height of the highest pixel of each identified particle.

Close examination of 3-D renderings of the images showed that the imaged particles were clearly round with no obvious faceting, so the assumption of spherical particles is valid. Prior to analysis of each image, the reference height of the substrate was set to zero at the mean point of the substrate’s histogram peak.

Results are presented in Table 1. Taking as a guide the AFM protocol for gold nanoparticles from the recent ASTM intercomparison⁶, the mean and standard deviation of particle heights derived from Figs. 3(b)–3(d) were discarded because they contained fewer than 100 particles. With the particle number density at the $\sim 5\mathrm{mm}$ location, it would have been necessary to increase the scan range to $9{\mathrm{\mu m}}^2$ or take another eight $1\mathrm{\mu m}\times 1\mathrm{\mu m}$ scan range images to image and measure at least 100 individual particles.

Table 1

Results from the series of images with particles in Fig. 3. The number of particles in Figs. 3(b)–3(d) was insufficient to determine statistically reliable values of mean and standard deviation.

Two measurements were derived from Figs. 3(e) and 3(f). The first measurement let the SPIP particle analysis function automatically identify particles and resulted in the inclusion of both individual particles and particle clusters. Whilst for the second measurement, particle clusters were excluded from the calculation so that only individual particles were included. With the relatively low level of agglomeration, there is little difference between the two sets of measurements.

The measured heights of the individual particles in the $\sim 5\text{-}\mathrm{mm}$ image and the $\sim 12\text{-}\mathrm{mm}$ image are plotted in Fig. 5. The figure clearly illustrates how measurement statistics improve with increased particle number density; however, these plots emphasize the necessity of imaging and measuring sufficient number of particles to achieve sufficient measurement statistics. Even with 125 individual particle measurements plotted in Fig. 5(b), there is neither an unambiguous Gaussian distribution nor an obvious peak to locate a mean particle size.

It should be noted that the images are based on a finite number of AFM scan lines. It is likely that the apex of each particle in an image does not exactly coincide with a scan line, so the exact value of $D$ for each particle has not been determined. An approach to resolve this problem could be to fit a model shape, such as an ellipsoidal cap, to the image of each particle to locate its apex.

Fig. 5

Plots of the particle height distributions: (a) for the image taken at the $\sim 5\mathrm{mm}$ position [Fig. 3(b)], the distribution is for only 12 individual particles, and (b) for the image taken at the $\sim 12\mathrm{mm}$ position [Fig. 3(f)], the distribution is for 125 individual particles.

Pixel height histograms of the images in Figs. 3(a), 3(e), and 3(f) are shown in Figs. 4(a), 4(b), and 4(c), respectively. The histogram in Fig. 4(a) is of the image at $\sim 4\mathrm{mm}$ with no visible particles and illustrates the flatness of the substrate as a Gaussian height distribution with a standard deviation of $\sim 0.22\mathrm{nm}$. In the histograms of images with visible particles (4(b) from the image at $\sim 10\mathrm{mm}$ and 4(c) from the image at $\sim 12\mathrm{mm}$), the overlaid red dotted line indicates the mean particle heights from Table 1 (the value including particle clusters), whereas the overlaid green dotted line indicates half the mean values which should correspond to the $h=D/2$ point in Fig. 2(b).

The mean particle height lines intersect the histograms between half and two-thirds the way up the trailing edge of the histogram in both histograms, and the half mean height lines cross in close proximity to the minimum point of the histogram between where the substrate peak dominates and where the particles dominate. This would suggest that for a set of particles with an approximately Gaussian distribution of heights, the mean height is approximately indicated by the half way point up the trailing edge of the histogram. The half mean height point occurs around a minimum rather than a maximum point as suggested by the model, which is probably due to the larger particles dominating the contribution at these points and to the added contribution of the lateral broadening of the particle images due to tip convolution.