# The Analysis of Periodic Order in Monolayers of Colloidal Superballs
## 4 Results and Discussion  
To elucidate the role of the particles' form factor on the structural analysis in (X-ray) diffraction patterns, monolayers of superball colloids with three different shapes, namely spherical, semi-cubic and cube-like, were prepared using the unidirectional rubbing method and investigated with SAXS. These colloidal monolayers were rotated around the rubbing direction (\(x\)-axis). First, we focus on 2D SAXS patterns and their analysis for a monolayer obtained from spherical colloids that can be described as a superball with a shape parameter of \(m=2\). Figures 3a-c show the SAXS patterns obtained for a monolayer prepared from solid colloidal silica spheres with \(D=1.10\,\mathrm{\SIUnitSymbolMicro m}\pm 3.2\%\). The three typical diffraction patterns imaged at rotations of \(\phi=0\), 20 and 40\({}^{\circ}\) all show distinct peaks that indicate the spheres have formed well-defined periodic structures in the monolayer. The presence of separate peaks, instead of a series of co-axial rings, indicate that the crystal orientation within the imaged area of the colloidal monolayer is fixed. This is a known result of the unidirectional rubbing method where the crystal grains align in the direction of rubbing [6,7]. Figure 3a shows the diffraction pattern imaged at a rotation of \(\phi=0^{\circ}\), where six-fold Bragg diffraction peaks indicate that the crystal structure of the colloidal monolayer is a two-dimensional close-packed hexagonal lattice. The peaks centered along the vertical direction of the images can be identified as the \(hk=10\) and 20 peaks of a hexagonal lattice, as indicated in Figure 3a. Upon rotating the sample around the \(x\)-axis, the Bragg peaks move in the vertical \(y\)-direction, which makes the diffraction pattern appear elongated (Figure 3b,c). In addition, it can be seen that the intensities of the peaks in the diffraction patterns change relative to each other. Specifically, the 10 peak is strongly visible in the pattern at \(\phi=0\) and \(20^{\circ}\), but disappears at \(\phi=40^{\circ}\), while the 20 peak is visible at \(\phi=0\)  
Figure 2: A schematic representation of the small-angle X-ray scattering (SAXS) setup. Here, the monolayers of colloidal superballs are rotated around the \(x\)-axis. The angle of rotation around this axis is denoted by \(\phi\). The incoming X-ray beam travels along the \(z\)-axis. The scattering patterns are imaged in \(x\),\(y\)-space. A beam-stop was used to prevent damage to the detector. Note that this schematic representation is not drawn to scale. and 40\({}^{\circ}\), but disappears at \(\phi=20^{\circ}\). By contrast, the 30 peak is only slightly visible at \(\phi=20^{\circ}\) and invisible at both \(\phi=0\) and 40\({}^{\circ}\). These trends can be more easily visualized using the cross sections of the scattering patterns in the vertical direction at \(q_{x}=0\) um. Our analysis revealed that upon rotating the sample the positions of the maxima in the structures shifted to a higher \(q_{y}\)-values. This shift is directly related to the rotation and by scaling the scattering vector \(q_{y}\) with a factor of \(\cos(\phi)\) all peaks in the various cross sections fell at equal values of \(q_{y}\text{cos}(\phi)\), as shown in Figure 3d for \(\phi=0\), 10, 20, 30 and 40\({}^{\circ}\). In this case we find maxima at \(q_{y}\text{cos}(\phi)=5.95\), 11.90 and 17.85 um\({}^{-1}\) for the 10, 20 and 30 peaks, respectively. Thus, by considering the cross sections of the diffraction patterns imaged at various rotations of \(\phi\), a more precise analysis can be performed to locate the positions of the various \(hk\) peaks. The strong changes in the intensity of the peaks observed upon rotating the sample are due to the presence of minima in the form factor. The distances of periodicity in the sample in the \(y\)-direction decreased upon rotation and thus correspond to higher \(q_{y}\)-values in reciprocal space.