356 | Nature | Vol 577 | 16 January 2020
Article
square (KCl)–hexagonal (pillars)–dodecagonal (AgCl) lattice (the SHD
lattice, see schematic in Fig. 1h).
Pillar templates were infilled by molten AgCl-KCl eutectic that
was directionally solidified at various rates (see Extended Data
Fig. 1b and Methods for details), leading to different values of g/λ (λ
determined outside the template region; see Extended Data Fig. 1c).
When g/λ ≤ 0.25, a disordered pattern (termed no-foil) persists (see
Extended Data Fig. 3a–d). Upon increasing the solidification rate such
that 0.4 < g/λ < 0.75, trefoil patterns were formed (see Extended Data
Fig. 3f–i). In the transition region when g/λ ≈ 0.3, a mixed, predomi-
nantly trefoil, structure is present (see Extended Data Fig. 3e). For
g/λ ≈ 0.95, quatrefoil patterns dominate (see Extended Data Fig. 4a,
b). When g/λ ≈ 1, the cinquefoil pattern appears (see Extended Data
Fig. 4c), and for g/λ ≥ 1.05, the hexafoil pattern is obtained (see Extended
Data Fig. 4d–f ). In Extended Data Fig. 5 we include quantitative image
analysis (see Methods for details) of the local and long-range order in
these patterns. The cross-sectional view of these spoke-like patterns
shows a tilted alignment of the eutectic phase boundaries (Extended
Data Fig. 6a). Thermal simulations based on our experimental setup
suggest that the solidification front within the bulk of the template
would be slanted (Extended Data Fig. 6b) owing to the nonplanar geom-
etry of the sample (see Methods for details). In experiments, there may
be other factors that affect the orientation of the solidification front,
including inhomogeneities in thermal properties. These factors are
ignored in the phase-field simulations, which assume a linear thermal
gradient parallel to the pillar axis, leading to a generally planar solidifi-
cation front perpendicular to the pillar axis; additionally, differences
in the thermal conductivities of the eutectic phases and pillars are not
considered (see Methods for details). Despite these simplifications,
the phase-field simulations (Fig. 2 ) produce structures that match the
experimentally observed structure of the eutectic (Fig. 1 ).
In phase-field simulations (see Methods for details), when g/λ = 0.163,
λ is substantially larger than the edge gap (g = 220 nm) of the template
and the template does not impose order on the eutectic pattern, result-
ing in a no-foil structure (see Fig. 2c, j). For increased solidification
rates, the simulations predict the trefoil pattern for 0.325 ≤ g/λ ≤ 0.65 1
(see Fig. 2d–f, k–m), the cinquefoil pattern for g/λ = 0.814 (see Fig. 2g,
n), and the hexafoil pattern for g/λ ≥ 0.976 (see Fig. 2h, i, o, p). We note
that phase-field simulations suggest that it is necessary to have the
solidification direction parallel to the template pillar axis (that is, along
the z axis; see Fig. 2a, b), rather than perpendicular to it, for these highly
ordered patterns to emerge. These results reveal that at the solidifica-
tion front within the template, the pillars disrupt the natural edgewise
diffusion of the lamellar eutectic by compelling the diffusion fields
to obey constraints set by the template geometry. To maintain the
requirement of consistent diffusion path lengths^17 within this modi-
fied diffusion field, the eutectic solidifies in spoke-like patterns while
preserving the overall hexagonal symmetry imposed by the template.
The patterns observed for various g/λ in experiments and phase-field
simulations are mapped in Fig. 3. This map suggests that certain ranges
of lamellar spacings and template edge gaps will result in a single type
of spoke-like pattern, a useful finding for setting the parameters to
achieve a specific mesostructure. While some experimentally observed
patterns were not observed in the simulations, this is probably due to
enforcement of periodic boundary conditions and the use of a domain
size of one template unit cell, which constrains the system and pre-
vents the emergence of asymmetric patterns or patterns with larger
periodicities. Simulations were also performed for g = 440 nm (see
Extended Data Fig. 7), showing the possibility of obtaining additional
patterns by changing the pillar diameter as well as g/λ. The effects of the
shape of the pillars (circular and oval) were also investigated via phase-
field modelling (see Extended Data Fig. 8). The eutectic maintains the
overall pattern expected for a hexagonal arrangement of pillars, but
the morphologies of individual phases are strongly dependent on the
exact shape of those pillars, showing the versatility of this process.
Building on these findings, template-directed eutectic solidification
was also explored within a monolayer silica colloidal crystal in which
spherical silica colloidal particles are assembled into a two-dimensional
hexagonally close-packed structure (see Fig. 4a, b, and Extended Data
Fig. 9a). The sphere diameter, d, was 560 nm (comparable to the pillars).
At a solidification rate where λ ≈ d, the eutectic pattern organized into
a trefoil pattern (see Fig. 4c), while a hexafoil pattern (see Fig. 4d) was
observed when λ < d. As expected, when the solidification rate was slow
such that λd≫ , the colloidal crystal template imposes no order on the
eutectic pattern, resulting in a disordered structure (see Extended Data
Fig. 9b). The trefoil pattern in this template resembles the ArchimedeanλFig. 1 | Selected microstructures formed by template-directed eutectic
solidification. a, SEM image of AgCl (bright)-KCl (dark) eutectic solidified at a
cooling rate of 22 °C min−1. Average λ as defined in the inset is 420 nm. The
solidification direction is out of the image (z axis), as indicated by the red
dotted circle. b, Plan-view SEM image of a pillar template sample showing the
hexagonal arrangement of pillars; here g = 220 nm, as defined in the inset.
c–h, SEM images of microstructures, with lattice schematics shown for
selected cases. c, Trefoil, e, quatrefoil, f, cinquefoil and g, hexafoil patterns
with 3, 4, 5 and 6 KCl spokes per unit cell of the template (see schematic in
Extended Data Fig. 2), respectively, were obtained by varying the solidification
conditions. d, Schematic of the Archimedean honeycomb lattice (for
c); h, schematic of the Archimedean SHD lattice (for g). Parts of the SEM images
(in c and e–g) are false-coloured, with AgCl as yellow, KCl as blue and Ni pillars
as black. Uncoloured SEM images are displayed in Extended Data
Figs. 3g, 4b, c, d. All scale bars, 1 μm.