Patterns Formed by Shadows of Spheres
Phase diagrams obtained via black CM2 1⁄4 0 and yellow CM3 1⁄4 0 curves for shadow overlap with nearest and next-nearest neighbors. Top left panel: Close-packed square lattice phase diagram (2r=a 1⁄4 2r=b 1⁄4 1, γ 1⁄4 π=2) reflects the fourfold lattice symmetry. Bottom left panel: Dense hexagonal lattice phase diagram (2r=a 1⁄4 2r=b 1⁄4 0.95, γ 1⁄4 π=3) with sixfold symmetry. [From S.V. Kostinski, E.R. Chen, and M.P. Brenner, "Characterization of Patterns Formed by Shadows of Spheres," Phys. Rev. Lett. 112, 235502 (2014) | DOI: http://dx.doi.org.ezp-prod1.hul.harvard.edu/10.1103/PhysRevLett.112.235502.]
Motivated by colloidal lithography, Harvard Physics Grad Student Sarah Kostinski, SEAS Postdoc Elizabeth Chen, and Professor Michael Brenner studied the problem of characterizing periodic planar patterns formed by shadows of spheres. The set of patterns accessible to shadow lithography spanned by lattice types, tilt, and rotation angles is rich, but topological considerations of shadow overlap along simplex edges and faces lead to just 4 þ 1 distinct categories. These planar patterns are in one-to-one correspondence with a 4-valued index linked to Cayley-Menger determinants. The characterization is confirmed by a phase diagram which predicts surface patterns for any experimental geometry.