Article
thermal conductivity of AgCl-KCl eutectic solid (κ = 3.25 W m−1 K−1)^22. A
cooling rate of 10 °C min−1 was enforced on the bottom surface of the
aluminium plate (that is, the cooling stage). Newton’s law of cooling
was applied to all other surfaces, with an air temperature of 300 K and
a thermal transfer coefficient of 10 W m−2 K−1 (ref.^38 ). The evolution of
the shape of the solidification front (approximated by the eutectic
temperature isocontour at 591.73 K) was calculated using element
sizes, ensuring a minimum of five elements across each feature. A
simulation was also performed with a eutectic thermal conductivity
corresponding to the liquid eutectic (κ = 0.45 W m−1 K−1)^22. The result
was qualitatively similar to that of κ = 3.25 W m−1 K−1 case, but the tilt of
the solidification front was more pronounced in the outer region of
the disk and there was almost no tilt in the inner region. While we find
that changing thermal conductivity has minimal impact on the shape
of the solidification front, we note that there are other simplifications
made in our model. First, the electrodeposited Ni pillars probably have a
lower thermal conductivity than pure Ni, and second, the alumina layer
will serve as a thermal barrier between the pillars and the eutectic (the
thermal conductivity of amorphous alumina grown by atomic layer
deposition is^39 ~2 W m−1 K−1, similar to the thermal conductivity of the
eutectic). This alumina layer may reduce thermal coupling between
the pillars and the solidifying eutectic. For simulations of monolayer
silica colloidal crystal templates, the substrate was instead assumed
to be a silicon layer (0.6 mm thick, κ = 130 W m−1 K−1). The bulk layer
was assumed to be a composite of silica colloidal crystal and eutectic
(560 nm thick, κ = 2.8 W m−1 K−1) with no overlayer.
Phase-field simulations
Solidification of the AgCl-KCl eutectic along the axis of the pillars was
simulated using a template-directed eutectic solidification model
based on the phase-field model developed by Folch and Plapp^40 and
as applied in other templated-eutectic cases^31 ,^32. Model parameters,
including material properties, were taken from refs.^22 ,^32. Simulations of
template-directed solidification were conducted over computational
domains representing a unit cell of the hexagonal lattice of pillars, with
dimensions of 780.6 nm and 1,352 nm and an edge gap of 220 nm. Peri-
odic boundary conditions were assumed at the unit-cell boundaries,
and no-flux boundary conditions were imposed at the template-eutec-
tic interfaces through the use of the smoothed boundary method^41. The
initial lamellar spacings were chosen such that their integer multiples
(1 to 7) would fit in the 1,352 nm domain width (see Fig. 2c–i), and the
solidification velocities were set such that they would result in these
lamellar spacings in the absence of a template. These initial lamellae
act as a solid seed in the simulations. The grid size of the computational
domain is set to provide sufficient resolution over the diffuse interface,
which must in turn be sufficiently smaller than the lamellar spacing,
to ensure numerical accuracy without unnecessary expenditure of
computational resources. Thus, the simulations with smaller lamellar
spacing have higher resolutions, resulting in the sharper appearance
of interfaces than those with larger spacing in Fig. 2. A linear thermal
gradient of 10^5 K m−1 was applied in the solidification direction. This
thermal gradient is probably larger than that found in the physical
system, but it allowed the simulation to reach a steady-state struc-
ture more quickly without affecting the final morphology. The linear
thermal gradient forced the solidification front to be perpendicular to
the axis of the pillars (that is, parallel to the x–y plane; see Fig. 2a, b).
The solidification of the eutectic was simulated along the axis of the
pillars (that is, along the z axis; see Fig. 2a, b) until a stable, steady-
state structure was attained. While this model does not explicitly
specify the contact angle, it closely approximates the model that
specifies a 90° contact angle on a flat surface of the template material,
corresponding to a system with both solid phases having the same
interfacial energy. This assumption of equal interfacial energy can be
justified by examining the experimental images, which show most of
the interfaces intersecting the pillar surface at approximately 90°.
Phase-field simulations were only performed for the solidification
direction parallel to the axis of the pillars because a tilted solidifica-
tion front requires a much larger computational domain. Phase-field
simulations were not performed for the monolayer colloidal-crystal
case because the level of resolution necessary to resolve the evolu-
tion in the small gaps between spheres made the simulation costly.Data availability
Data that support the findings of this study are available within the
paper, and from the corresponding author on reasonable request.Code availability
The phase-field simulations were performed using a custom code writ-
ten in Fortran 90. The code is available from K.T. ([email protected])
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Acknowledgements This work was supported through the Air Force Office of Scientific
Research Multidisciplinary University Research Initiative FA9550-12-1-0471. This work was
carried out in part in the Materials Research Laboratory Central Facilities, and in the Beckman
Institute for Advanced Science and Technology, University of Illinois. E.H. and K.T.
acknowledge use of the computational resources in the Department of Defense High-
Performance Computing Modernization Program. We thank J. Kohanek for suggestions and
discussions on solidification in pillar templates, Z. Ou and Q. Chen for help with image
analysis, G. Huang for help with thermal profile simulations, and K. Tyler and V. Mohan for
critical feedback on the manuscript.Author contributions A.A.K. and E.H., under the supervision of P.V.B. and K.T., respectively,
designed the study. A.A.K. and R.Z. fabricated the templates. A.A.K. performed solidification
experiments and sample characterization. E.H. performed the simulations. A.A.K., E.H. and R.Z.
analysed the data. All authors prepared the manuscript.Competing interests The authors declare no competing interests.Additional information
Correspondence and requests for materials should be addressed to P.V.B.
Peer review information Nature thanks Elizabeth Dickey, Alain Karma and the other,
anonymous, reviewer(s) for their contribution to the peer review of this work.
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