# The Analysis of Periodic Order in Monolayers of Colloidal Superballs
## 5 Conclusions  
Monolayers of differently shaped silica superballs, namely spherical, semi-cubic and cube-like, were prepared using the unidirectional rubbing method and analyzed with SAXS.  
Figure 7: (**a–c**) The deduced structure factor \(S(\vec{q})\) in reciprocal space and (**d–f**) the corresponding lattices in real space for the (**a**,**d**) solid spherical, (**b**,**e**) hollow semi-cubic and (**c**,**f**) hollow cube-like superballs. Here \(\vec{b}_{1}\) and \(\vec{b}_{2}\) are the primitive reciprocal basis vectors, separated by angle \(\beta\), and \(\vec{a}_{1}\) and \(\vec{a}_{2}\), separated by angle \(\alpha\), are the corresponding primitive lattice vectors. For the cube-like superballs we have instead defined \(\vec{b}_{2x}\) as the horizontal component that is perpendicular to \(\vec{b}_{1}\). The Bragg peaks are labelled on their left. The direction of the mechanical rubbing is indicated in the images using a red arrow.  
In the 2D SAXS patterns some of the Bragg peaks appeared to be missing due to the presence of minima in the particle form factors. Due to the anisotropic shape and the hollow nature of the particles, it was difficult to predict which peaks are missing, which complicated the analysis of the patterns. By imaging the diffraction patterns of the monolayers over a range of rotations around the \(x\)-axis, the missing Bragg peaks were visualized. By doing so, the structural analysis was possible for all three superball shapes. We found that densely packed monolayer structures were obtained for all three types of particles using the unidirectional rubbing method. The spherical superballs formed hexagonal lattices with a high degree of positional order in all directions, whereas upon increasing the shape parameter \(m\) it was observed that this high degree of positional order in the direction of rubbing remained, but weakened in the direction perpendicular to the direction of rubbing, resulting in the formation of the sliding phase for the cube-like superballs. In the future, we will use this analysis method to further study and characterize the lattices of colloidal superballs assembled using the unidirectional rubbing method.  
Thus, by exploring the three-dimensional reciprocal space of a two-dimensional sample we were able to effectively exclude the influence of the anisotropic form factor and analyze the densely packed structures of the colloidal monolayers in greater detail. As a final remark, we note that the problems discussed in the present work are not X-ray specific, but can as well be of importance for small-angle neutron scattering studies. If samples possess periodic ordering in fewer dimensions, in 1D or 2D, a rotation of the sample may allow the revelation of all essential diffraction features even in the case of Bragg peaks coinciding with form factor minima.  
Conceptualization, D.N.t.N. and A.V.P.; methodology, D.N.t.N. and A.V.P.; software, D.N.t.N.; validation, D.N.t.N., J.-M.M. and A.V.P.; formal analysis, D.N.t.N.; investigation, D.N.t.N. and A.V.P.; writing--original draft preparation, D.N.t.N.; writing--review and editing, D.N.t.N., J.-M.M. and A.V.P.; visualization, D.N.t.N.; supervision, J.-M.M. and A.V.P.; project administration, A.V.P.; funding acquisition, A.V.P. All authors have read and agreed to the published version of the manuscript. This research was funded by NWO grant number 712.015.003 (D.N.t.N), 016.Veni.192.119 (J.-M.M.) and 195.068.1113 (ESRF beamtime).  
Not applicable.  
The data presented in the study are available on request from the corresponding author.  
[PERSON] is thanked for her help with the preparation of the spherical silica colloids. [PERSON] is thanked for his help at the beamline of the ESRF. [PERSON] and [PERSON] are thanked for providing technical support at the beamline. [PERSON] is thanked for useful discussions. The Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) is thanked for providing the financial support.  
The authors declare no conflict of interest.