# The Analysis of Periodic Order in Monolayers of Colloidal Superballs
## 4 Results and Discussion
Now that we have assigned all identifiable Bragg peaks in the diffraction patterns of the three superball shapes, we can begin to deduce their structure factors \(S(\widetilde{q})\) and determine what these superball lattices look like in real space. From the positions of the 10, 20 and 30 peaks we can determine the length of the first primitive reciprocal basis vector, \(\widetilde{b}_{1}\), according to Equation 5. For the spherical and semi-cubic superballs, we can determine the length of the second primitive reciprocal basis vector, \(\widetilde{b}_{2}\), from the positions of the 01, 11 and 21 peaks. Since for the cube-like superballs the 01, 11, 21 peaks appear to form a continuous band, we can instead define \(\widetilde{b}_{2x}\) as the horizontal component of \(\widetilde{b}_{2}\). Here, \(\widetilde{b}_{2x}\) is perpendicular to \(\widetilde{b}_{1}\). The deduced structure factors for the spherical, semi-cubic and cube-like superballs along with the identified Bragg peaks and their primitive reciprocal basis vectors are depicted in Figure 7a-c, respectively. From the primitive reciprocal basis vectors \(\widetilde{b}_{1}\) and \(\widetilde{b}_{2}\), we can extract the real space primitive lattice vectors \(\widetilde{a}_{1}\) and \(\widetilde{a}_{2}\) and, in the case of the cubic-like superballs, \(\widetilde{a}_{2x}\). Their corresponding real space lattices are shown in Figure 7d-f. For the spherical superballs we determined the length of the primitive lattice vectors to be \(a_{1}=1.21\pm 0.013\,\mathrm{\SIUnitSymbolMicro m}\) and \(a_{2}=1.19\pm 0.004\,\mathrm{\SIUnitSymbolMicro m}\), with an angle of \(\alpha=119.4\pm 0.26^{\circ}\). This confirms that these superballs indeed form a hexagonal lattice in real space with a well-defined and equal periodicity in each direction. For the semi-cubic superballs we found that \(a_{1}=1.11\pm 0.019\,\mathrm{\SIUnitSymbolMicro m}\) and \(a_{2}=1.04\pm 0.011\,\mathrm{\SIUnitSymbolMicro m}\), separated by an  
Figure 6: Numerically calculated form factors of the (**a**) solid spherical, (**b**) hollow semi-cubic and (**c**) hollow cube-like superballs at a rotation of \(\phi=0^{\circ}\). The orientation of the superball particles is depicted in the insets of the images. Here, the intensity is limited to 4 orders of magnitude, which is approximately the same range the detector used in these experiments is able to measure. angle of \(\alpha=117.8\pm 0.47^{\circ}\). Here, since \(a_{1}\) is significantly larger than \(a_{2}\), these semi-cubic superballs form a distorted hexagonal (\(\Lambda_{0}\)) lattice as expected for the densest packing of superdisks in 2D [28]. From \(a_{1}>a_{2}\) we further conclude that these semi-cubic superballs are oriented with their flat faces in the horizontal and vertical directions. For the cubic-like superballs that form a more cubic-like lattice we find that \(a_{1}=1.05\pm 0.027\,\mathrm{\SIUnitSymbolMicro m}\) and \(a_{2x}=1.08\pm 0.044\,\mathrm{\SIUnitSymbolMicro m}\). Furthermore, from the presence of broad and continuous \(h1\) and \(h2\) bands in \(S(\vec{q})\), we further conclude that these cube-like colloidal superballs form chains in the horizontal direction that are able to slide freely with respect to each other. Thus, upon increasing the shape parameter \(m\) of the superballs, several structural changes are observed in their lattice structures. First, the superballs have formed chains in the horizontal direction in the monolayers, which are aligned to the direction of the mechanical rubbing. Upon increasing the shape parameter \(m\), the ratio between \(a_{1}\) and \(a_{2}\) increases, which indicates that the spacing between the chains increases, as expected for superballs [28]. This leads to the ability for the chains to slide along each other for the most cubic-like superballs.