Statistical assessment of order within systems of nanoparticles: Determining the efficacy of patterned substrates to facilitate ordering within nanoparticle monolayers fabricated through electrophoretic deposition

The degree of order within nanoparticle monolayers deposited through electrophoretic deposition on lithographically patterned and unpatterned substrates was analyzed using four complementary measures of order: Voronoi-cell edge-fraction entropy, local bond-orientation order parameter, translational order parameter, and anisotropy order parameter. From these measures of order, we determined that the pattern had an influence on some aspects of the ordering within the nanoparticle monolayer but had no effect on others. The Voronoi-cell edge-fraction entropy did not measurably change due to the pattern, indicating that the pattern has no effect on the number of defects present. The translational order parameter also had no change due to the pattern. The local bond-orientation order parameter had a measurable change, indicating the pattern increased the bond ordering slightly. Also, the anisotropy order parameter developed herein indicated an increase in order. The direction of the increased order corresponded with the direction of the anisotropy designed on the patterned substrate, strongly suggesting that the pattern drives the particles to become more ordered.

Figure 1

(a) SEM image of iron oxide NPs on unpatterned region of substrate. (b) A segmented image of (a), with black areas representing voids in the film. (c) A processed version of (a), optimized for identification of particle locations. Identified particles are marked by red (gray) plus signs. (d)–(f) Corresponding images for the patterned substrates. (e) In this example, black regions represent both voids and pattern elements. (Scale bar $=100$ nm.)

Figure 2

(a) and (b) Voronoi tessellation, derived from an image of a nanoparticle film shown in Figs. 1c and 1f. Voronoi cells with one or more vertices within a black region of the segmented images [Figs. 1b and 1e] were removed. Cells were color coded based on the number of sides of the cell. No cells with greater than 8 or less than 4 sides were shown. (c) Histogram showing the fraction of particles with $n$ sides, where $n=3-9$. Within error, the unpatterned and patterned substrates were equivalent. (Scale bar $=$ 100 nm.)

Figure 3

(a) Distribution of ${\psi }_{6,\mathrm{local}}$ values for patterned and unpatterned regions. The patterned substrate distribution was shifted slightly to the right, indicating a greater degree of local ordering. (b) Pair correlation function which is used to calculate $\tau$, the translational order parameter.

Figure 4

The properties of ${\Psi }_{\mathrm{an}}$ as determined by simulating a set of bond angles (a) The blue (dark) line demonstrates how ${\Psi }_{\mathrm{an}}$ varies as the standard deviation of bond angles varies. The standard deviation is plotted in percent of total angular range. The red (light) line shows the derivative of ${\Psi }_{\mathrm{an}}$ as a function of the standard deviation, indicating the sensitivity of ${\Psi }_{\mathrm{an}}$ to changes in the standard deviation. The inset is an example of simulated bond angles for a set of bond angles with a 0.001$\%$ standard deviation and an angular range of 2π. (b) This graph demonstrates that even for completely disordered systems, ${\Psi }_{\mathrm{an}}$ will not be zero and will depend on the sample size. The sample size is a measure of the number of simulated correlated regions. Each correlated region is randomly oriented, and all bonds within a single correlated region are equivalent. ${\Psi }_{\mathrm{an}}$ is calculated 100 000 at each simulated sample size. The $+$1σ, $+$2σ, and $+$3σ lines indicate the mean value of ${\Psi }_{\mathrm{an}}$ plus 1, 2, and 3 times the standard deviation of the 100 000 calculated values. The red (light gray), individual dot represents the value measured for the unpatterned substrate. The dot lies below the mean ${\Psi }_{\mathrm{an}}$, which is expected since ${\Psi }_{\mathrm{an}}$ is simulated for perfect hexagons. The black dotted line lies along the value measured for the patterned substrate. The sample size for the patterned substrate is well beyond the edge of the simulation at 2180. Thus, the patterned substrate is well above the $+$3σ line. The gray region (top) is forbidden because mathematically, ${\Psi }_{\mathrm{an}}$ cannot exceed 1. The inset shows a histogram of 100 000 ${\Psi }_{\mathrm{an}}$ values simulated for a sample size of 600.

Figure 5

A subset of the Voronoi cells from Fig. 2 having six sides and all edge lengths within 60$\%$ of the median edge length for (a) unpatterned regions and (b) patterned regions. The color indicates the circular mean (over the range 0°–60°) of the six bond angles for each cell.

Figure 6

The black (square data points) line is the measured number of bonds that are oriented within an angular range, measured relative to the $x$ axis of one of the SEM images of a patterned substrate. The red (light), solid line is a von Mises (circular analog to a normal distribution) fit to the data. Based on the fit, the peak in the distribution occurs at 22.8°±1.3°, agreeing with the direction of the patterned elements as well as the angle measured by ${\Psi }_{\mathrm{an}}$. The blue line (lower line with circle data points) shows the equivalent bond-orientation data for the unpatterned substrate.