Characterization of Patterns Formed by Shadows of Spheres

Motivated by colloidal lithography, we study 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 us to just $4+1$ 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.

Figure 1

(a) Schematic of the shadow lithography process (side view): A flat substrate is masked by a monolayer colloidal crystal of spheres as evaporated metal atoms are deposited in the direction of the orange rays oriented at tilt angle $\varphi$ from the normal. The substrate can be rotated by angle $\theta$. (b) Top-down view of patterns obtained by changing rotation ($\theta$) and tilt ($\varphi$) angles with a 0.9 sphere diameter-to-lattice spacing ratio in a square lattice. Shadow regions are gray, while pattern hue and brightness indicate $\theta$ and $\varphi$ values, respectively. See Supplemental Material [24] for a close-up view and additional figures.

Figure 2

Top-down views of the four topologically distinct pattern categories (black and white insets) which are determined by shadow overlap and analyzed by tiling the plane in simplices with vertices at the centers of three neighboring elliptical shadows. Here we show triangles for a sphere with pairs of nearest neighbors in a hexagonal lattice. Edges with shadow overlap are green; those without are red. Possible 3-ellipse overlap topologies show that the number of triangle sides (0, 1, 2, or 3) where overlap of shadows occurs uniquely determines the four pattern categories: (a) The “fully connected” pattern has zero triangle sides with shadow overlap; (b) the “banded-connected” pattern has one triangle side with shadow overlap; (c) the “disjoint bar” pattern has two such sides; (d) the “disjoint triangle” pattern has shadow overlap occurring on all three sides. The latter case (triangle pattern) can be distinguished from (e) the absence of pattern (“total eclipse”) by examination of three-shadow area overlap along the triangle face (i.e., if the union of the three shadows entirely spans the triangle interior).

Figure 3

Lattice geometry. (a) Top view: Blue colloidal spheres are arranged in a lattice defined by vectors $\mathit{a}$ and $\mathit{b}$, masking the substrate from metal deposition. The trajectories of deposited atoms project 2 D vector $\mathit{p}$ at rotation angle $\theta$ with respect to lattice vector $\mathit{a}$. (b) Top view of an example metal pattern (in blue) after removal of the colloidal mask. The shadows (gray region) projected by the spheres are ellipses, whose centers are also separated by lattice vectors $\mathit{a}$ and $\mathit{b}$.

Figure 4

Shadow overlap analysis. (a) Compression: All shadow ellipses’ major axes are parallel; a spatial compression by factor $\cos \varphi$ along this axis transforms elliptical shadows into circles of radius $r$ while preserving the shadow overlap topology. (b)–(d) Shadow overlap along edges: Two circular shadows $i$ and $j$ overlap along the edge which connects their centers if a 2D simplex (triangle) with edge lengths $\{r,r,d_{ij}\}$ can be formed; i.e., $r$ must be larger than $d_{ij}/2$. (c) shows the crossover case ($C{M}_2=0$) which separates the overlap case (b) from case (d). (e)–(g) Interior shadow overlap: The union of three circular shadows $i$, $j$, $k$ completely spans the interior of an acute triangle whose vertices are their centers if a 3D simplex (tetrahedron) with edge lengths $\{r,r,r,d_{ij},d_{jk},d_{ki}\}$ can be formed, i.e., if $r$ is larger than the triangle’s circumradius. (f) shows the crossover case ($C{M}_3=0$) between overlap (e) and no overlap (g), in which all three shadows intersect at a point.

Figure 5

Phase diagrams obtained via black $C{M}_2=0$ and yellow $C{M}_3=0$ curves for shadow overlap with nearest and next-nearest neighbors. Left panel: Close-packed square lattice phase diagram ($2r/a=2r/b=1$, $\gamma =\pi /2$) reflects the fourfold lattice symmetry. Right panel: Dense hexagonal lattice phase diagram ($2r/a=2r/b=0.95$, $\gamma =\pi /3$) with sixfold symmetry.