Perfection in Nucleation and Growth of Blue-Phase Single Crystals

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Surfaces, Interfaces, and Applications

Perfection in Nucleation and Growth of Blue Phase Single Crystals: Small Free Energy Required to Self-Assemble at Specific Lattice Orientation Xiao Li, José A. Martínez-González, Kangho Park, Cecilia Yu, Ye Zhou, Juan J. de Pablo, and Paul F Nealey ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b18078 • Publication Date (Web): 14 Feb 2019 Downloaded from http://pubs.acs.org on February 14, 2019

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ACS Applied Materials & Interfaces

Perfection in Nucleation and Growth of Blue Phase Single Crystals: Small Free Energy Required to Self-Assemble at Specific Lattice Orientation Xiao Li1, 2*, Jose A. Martinez-Gonzalez,3*, Kangho Park1, 4, Cecilia Yu1, Ye Zhou1, Juan J. de Pablo1,2† and Paul Nealey1, 2†, 1Institute

for Molecular Engineering, The University of Chicago, Chicago, IL 60637.

2Material

Science Division, Argonne National Laboratory, Lemont, IL 60439.

3Facultad

de Ciencias, Universidad Autónoma de San Luis Potosí, Lateral Av. Salvador Nava s/n, San Luis Potosí 78290, SLP, México. 4Department

of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, South Korea. *These

authors contributed equally to this work.

†Correspondence

to: [email protected], [email protected]

Abstract Chemically patterned surfaces can be used to selectively stabilize blue-phases as macroscopic single crystals with a prescribed lattice orientation. By tailoring the interfacial free energy through the pattern characteristics, it is possible to set, with nanoscale precision, the optimal conditions to induce a spontaneously blue-phase crystal nucleation on the patterned substrate where a uniform, defect-free, blue-phase single crystal is finally formed in matter of seconds. The chemical patterns took into consideration in this work are made up of alternated stripe-like regions of homeotropic and planar anchoring. By varying the stripe pattern dimension, including the period and ratio of the planar/homeotropic anchoring width, it is possible to generate blue phase I-single crystals with 1

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(110) lattice orientation, and blue phase II-single crystals with either the (100), (110) and (111) lattice orientation. Continuum mean field calculations of the studied systems serve to explain, in terms of the free energy of the systems, how the pattern dimensions favor certain crystallographic orientation while penalize the others. We found that a small free energy difference is sufficient to drive the nucleation and growth of blue phases into a certain lattice orientation. Therefore, a processing window for obtaining arbitrary large blue phase-single crystals with pre-designed lattice orientation, highly aligned reflective peaks and significantly short forming time is provided here, which is essential for manufacturing and modulating of optical devices and photonics.

Keywords: blue-phase liquid crystal; chemically patterned surfaces; interfacial free energy; lattice orientation; single crystal

Introduction Blue phases (BPs) of chiral liquid crystals have attracted significant research interest because they possess unique self-assembled three-dimensional (3D) crystalline structure in soft matter, and advanced performance as fast electro-optical devices, photonic crystals, stimuli response sensors and three-dimensional lasers1-9 to name a few. In our previous work, we have successfully used these materials to study the liquid analog of some characteristic phenomena involved in atomic crystals, such as the diffusion-less martensitic-like transformation epitaxial growth

10, 11.

10,

crystal nucleation and

A remarkable difference of a BP-crystal with respect to an atomic one, is

that the first one is formed by submicron-size unit cells that consists of billions of molecules in a liquid-crystalline state. Therefore, it is noteworthy to mention that chemical patterns can be applied on BP molecules, through directed self-assembly process, to ultimately produce a macroscopic single BP-crystals free of grain boundaries. This is possible only if the pattern possesses the proper symmetry and interfacial anchoring. In our previous work11, we explain how mean-field theoretic calculations can be used, first to design and then to optimize, binary-anchored patterns that will favor the crystalline nucleation of a given BP, this led to different pattern designs each one conceived to favor a specific crystallographic orientation. In this work, we show that a simple pattern symmetry can be used to favor a variety of crystallographic orientations if we understand how the pattern dimensions affects the free energy of the BP according to its lattice orientation.

2


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BP molecules can self-organized locally into structures called double twist cylinder (DTC). The DTCs cannot fill into the entire volume of the system, but spontaneously assembled into cubic crystalline symmetries with 3D long range order12,13. There are two types of cubic BPs as the temperature increases from the cholesteric (Chol) phase: the so-called BPI which possess a bodycentered cubic (BCC) symmetry and the BPII with a simple cubic (SC) symmetry, in both cases the lattices constants are in the submicron scale14. Controlling the BP-lattice orientation has been extremely difficult to achieve since, unlike nematics, the director is not uniform. The polycrystalline and multi-platelet domains presented in the BP specimens inevitably deteriorate their optical response; additionally, these BP samples show multiple Bragg reflection intensities resulting from different crystallographic planes and their orientations. Therefore, to fully exploit the optical properties of BPs it is desired to have macroscopic BP crystals. Past studies show that through conventional surface rubbing method along with electrical field and/or thermal treatments, a large area monodomain-like liquid crystalline BPs can be obtained over the whole fabricated surface 15-17. A recent study shows an efficient temperature-gradient process to grow BP-single crystals of macroscopic (~ cm) dimenssions18. However, in the previous techniques, a BP platelet takes several hours to grow a few micrometers which favors the same BP-lattice orientation. As we mentioned before, binary homeotropic/planar patterns are designed using a mean free energy model of the confined blue-phase liquid crystal11. Through theoretical calculations, the dimensions of the homeotropic and planar regions are optimized in terms of the behavior of the free energy difference, this allows to identify the conditions where a BP with a given (hkl)-lattice orientation can be favored. The optimized pattern yielded the maximum free energy difference between the desired lattice orientation and all other potential orientations. Here, our efforts are motivated by

investigating further the mechanism by which a small free energy difference can be sufficient to determine the formation of BP single crystals with particular lattice orientations, as well as the advent of directed self-assembly of block copolymers where chemical patterns can induce the copolymers to adopt complex structures that differ from their morphologies in bulk 19, 20. For the BPs on the engineered patterned boundary conditions, according to the pattern parameters it is possible to estimate the free energy difference between BPs that differs on the lattice orientation. In this way, we can set up the conditions to promote a spontaneously crystal nucleation where BPsingle crystals, with a certain phase and lattice orientation, free of dislocations and grain 3



boundaries, can be formed in matter of seconds. Therefore, we build on past work10, 11 to analyze the process window of epitaxial BPs as an arbitrary large single crystal, i.e., the range of periodicity or width ratio of the stripe-like pattern that stabilizes the formation of uniform BPI and BPII with pre-designed lattice orientation. We focus our study on various phase morphologies and lattice orientations of BPs on such stripe-like chemical patterns. Specifically, we analyze parameters such as the chemical pattern periodicity on the substrate, Ls, its relation to BP’s natural period known as unit cell size (aBPII), the dimension of the chemical pattern, W, and the total free energy, including the elastic energy of the BPs and the interfacial energy between the BPs and the chemically patterned substrate. Unveiling that how these parameters affect the equilibrated morphologies of BPs will be critical for both further improving the current technologies and expanding the spectrum of directed self-assembly BPs applications.

Results and Discussion The chemically patterned surfaces used for this work were prepared following a technique that was recently developed in our group.21 Specifically, the hybrid cell used in this work comprises a modified Octadecyltrichlorosilane (OTS) glass surface as top surface imposing homeotropic anchoring, and a homeotropic bottom surface including a 440 m  440 m chemical pattern region with alternating planar and homeotropic anchoring stripe. Figure 1a illustrates schematically the fabrication process of the chemically nanopatterned surfaces, the pattern parameters of interest, and a 3.5 µm thick hybrid anchoring LC cell. PMMAZO brush was uniformly deposited on the silicon substrate. Photoresist was subsequently coated and then patterned on top of the brush layer using e-beam lithography to open up arrays of stripe trenches with a period of Ls. Oxygen plasma was used to remove the brushes in areas that were not protected by the overlying photoresist. After removing the photoresist, stripes regions of planar anchoring were created in a homeotropic anchoring background21, 22. The period of the alternative planarhomeotropic anchoring stripes, Ls is given by 𝐿𝑆 = 𝛿 𝐿0, where L0 is a reference pattern periodicity that we took from the value of the BPII lattice constant. Ls is systematically varied from 130 nm to 450 nm. For each case, the width of the planar-anchoring stripes, Wp, is related to the referred periodicity and either kept as 75 nm or systematically varied from 65 nm to 225 nm; while the width of the homeotropic-anchoring strips is defined as WH =Ls - Wp. The chiral liquid crystal used 4


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in this work (Fig. 1b) consisted of MLC 2142 mesogens mixture with 36.32 wt% of the chiral dopant 4-(1-methylheptyloxycarbonyl)phenyl-4-hexyloxybenzoate (S-811). Consistent with previous works10, 11, the blue phases of this chiral LC are characterized by unit cell sizes of BPI as aBPI = 255 nm and BPII as aBPII = 150 nm, respectively. The phase transition temperatures of such confined BPLC materials were measured to be: Chol to BPI at 39.5 ± 0.1 oC; BPI to BPII at 40.5 ± 0.1 oC; BPII to Iso at 42.5 ± 0.1 oC. In this work, BPI(hkl) and BPII(hkl) are used to represent a given BP for which the (hkl)-lattice planes are perpendicular to the incident light. In the BPII, there is a four-arm junction disclination line at the center of the unit cell (Fig. 1b). Since the symmetry of the BPII is simple cubic therefore light can be possibly reflected from all the crystallographic planes (h k l). For BPI, however, because of its BCC symmetry, light can be reflected from planes when the sum of the Miller indices (h + k + l) is an even number. According to Braggs’ law, when light irradiate normal to the (h k l) plane, the reflected wavelength is given by 𝜆(ℎ𝑘𝑙) = 2𝑛𝑎 ℎ2 + 𝑘2 + 𝑙2, where n is the refraction index and a is the BP-lattice constant. Considering that the lattice constant of the obtained BPII is 150 nm, only the (100) planes reflects visible light with 𝜆(100) = 450nm, resulting in a uniform blue color (Fig. 1b). For BPI, the lattice constant is a=255 nm and the (200) and (110) planes reflect visible light with 𝜆(200) = 382.5 nm (dark blue) and 𝜆(110) = 541 nm (green, Fig. 1b), respectively. Different than previous works10, 11,

for both cases showing in Figure 1b, simple stripe-like patterns with different pattern

dimensions are applied to stabilize a single crystal assembly of BPI(110) at 40.1 oC and BPII(100) at 41.7 oC. Based on our past theoretical predictions, we first focus on the alternating anchoring stripe pattern with Ls=L0, i.e., when the pattern period is the same as aBPII, and we systematically increase Wp from 0 to L0 (Figure 2a-b). Figure 2b shows when Ls= aBPII =150 nm, the free energy density difference (f) between the confined BPII and its value in bulk varies by changing Wp/Ls. In order to lower the free energy, the BP adopts a (100)-lattice orientation when the planar stripe ratio is between 0.2-0.7, while (111) becomes the preferred lattice orientation for uniform planar and homeotropic anchoring. In order to validate these predictions, we perform experiments starting from the Chol phase and using a step heating rate of 0.2oC/1min, i.e. increasing the temperature 5



by 0.2oC every 1 min. We found that the BPI is observed within the temperature regime from 39.5± 0.1 oC to 40.5± 0.1 oC; while the BPII is observed from 40.6± 0.1 oC to 42.8± 0.1 oC. Figure 2c shows the BPI and BPII obtained at 40.1oC and 41.7oC, respectively. We found that in the BPII temperature-region a homogeneous single-crystal of BPII(100) is formed on the chemical stripe-like patterns when 0.3L0 ≤ Wp ≤0.6L0 (Fig. 2C2-2C4). While in the BPI temperature regime, a polycrystalline texture dominated by platelets of BPI(110) and BPI(200) is produced, appearing as green and dark blue domains. For Wp=0.67L0, depending on the Wp/Ls ratio, the patterned surface can favor the (100) lattice plane (Fig. 2b) or produce a polycrystalline BPII texture with the (100) lattice plane as the main orientation (Fig. 2C5). We observe, that the multiple dark spots of the polycrystalline BPII do not disappear due to the kinetic hindrance to growth. As expected, for the limit cases, when Wp goes to 0 (uniform homeotropic anchoring) or when Wp goes to L0 (uniform planar anchoring), both BPI and BPII show a polycrystalline behavior. The above results confirm our previous predictions obtained by theoretic mean-field Landau-de Gennes calculations11, some representative simulations show the detailed disclination line configurations close to the pattern surface in Fig. 2d. It is essential to note that the overall behavior of the free energy is primarily affected by the strain induced by the different pattern morphologies, and even small free energy differences are sufficient to affect the BP crystal nucleation and growth. As a result, a given ratio of planar anchoring favors a specific BP-lattice orientation since it induces a local deformation preventing the BP material to adopt other, unfavorable configurations (Figure 2d). Based on these theoretical results and the obtained experimental evidence, it is a rational inference that the patterned stripe surfaces will stabilize a BPII(100) lattice plane with the ratio of planar stripe between 0.3L0 and 0.7L0. The optimal conditions to produce monocrystalline BPII with (100) lattice orientation correspond to a pattern periodicity equals to the blue phase unit cell size (150 nm) and Wp/Ls ~0.5 (Wp/WH ~ 1.0). Under these conditions, the molecular orientation of the BPII(100) at the surface is well adapted to the pattern symmetry as shown by the director field in Fig. 2d, leading to the formation of monocrystalline domains. The disclination lines are visualized in blue and correspond to iso-surfaces of the nemaitc order parameter, S (with S=0.35). Once we determine that the optimal pattern parameters that promotes a BPII(100)-single crystal correspond to a stripe pattern period equal to the BPII lattice constant, and a ratio Wp/Ls=0.5, the 6


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next step is to explore pattern periods where such a WP/Ls ratio is still useful. We then performed experiments where the pattern period changed from 130 nm to 450 nm. As shown in Figure 3a, when the pattern period is 130 nm-170 nm, which are values close to the BPII unit cell size, we observe the formation of a single-crystalline domain of BPII(100) with its characteristic uniform blue color. For the BPI temperature range, multiple small domains with (200) and (110) lattice orientations grow and produce a polycrystalline texture that reflects green and dark blue light. Once the stripe pattern period (Ls) is large enough (above 180 nm), within the BPII temperature range, the reflected light is no longer blue but appears dark. The dark color on pattern region indicates that for Ls ≥ 180 nm, either the (110) or (111) could be the possible BP-lattice orientations as proven by the corresponding Kossel diagrams. Here, it is important to highlight that we have found based on alternate stripe pattern surface, when WP/Ls ratio is fixed, the pattern period (Ls) determines the nucleation and grow of a BPII single crystal with either (100), (110) or (111) lattice orientation over the entire patterned area. Surprisingly, within the BPI temperature range the (110)lattice orientation is stabilized uniformly during the Ls=180 nm-200 nm, it appears as a single crystal with uniform green color. The corresponding Kossel diagram confirms the BPI(110) crystal structure. When the pattern period increases as integer times of the unit cell size aBPII, for instance, 300 nm (2.0L0) and 450 nm (3.0L0), a large area monodomain of BPI(110) is found to be dominant with some kinetic trapped (200) small domains inside. A similar trend is found when the pattern period is kept between 130 nm and 450 nm, and the planar anchoring stripe is fixed as 75 nm (Figure 3b). The difference lies when the pattern period is 300 nm (2L0): the BPII(100) grows as single crystal, however, the growth rate is significantly lower than the previous cases where the BPII(100) is formed within 1 min in all the patterned area, and the regions with different lattice orientations still remain after 5 min. For the pattern period of 3L0, a mixture of BPII(100) and BPII(111) is formed on the patterned area. The same situation happens during the BPI temperature range, the BPI forms a large area of monodomain crystal with (110) lattice orientation mixture with small domains of (200) lattice orientation, which shows dark blue color on the pattern surface when Ls=2L0 and 3L0. Similar to the previous cases, the kinetic hindrance of the BP-crystal growth keeps the material trapped in a polycrystalline behavior. Our experimental results demonstrate that a binary-anchored stripe like pattern, can promote the 7



assembly of single crystals of BPI and BPII with a wide variety of lattice orientations. Specifically, we have provided the pattern parameters to obtain single crystals of BPI(200), BPII(100), BPII(110) and BPII(111). At this point we turn to Landau-de Gennes calculations to do the spot check of those pattern periods and give additional information about the pattern parameters that will favor a specific BP-lattice orientation. Figure 4a shows when Ls=200 nm, the free energy density difference (f) between the confined BPII and its value in the bulk varies by changing Wp/Ls. The theoretical results show that starting from Wp/Ls~0.4 up to Wp/Ls~0.8, it is expected to obtain a BPII(110) single-crystal since its free energy is lower than the other two lattice orientations. From the experimental observation, when Ls=200 nm, Wp/Ls=0.5 (Fig. 3a) and Wp/Ls=0.375 (Fig. 3b) a single BPII(110) crystal was formed, which means that a small free energy difference is sufficient to drives the nucleation and growth of certain lattice orientation. Figure 4b shows the corresponding free energy density behavior associated to the BPI and Ls=180 nm; such results are in agreement with our experimental observations where the (110) lattice orientation is the preferred one for the BPI over the patterned region. These results are also consistent with previous experimental observation for BPs confined on slabs with uniform homeotropic or planar anchoring, where the BPI(110) is found to be the preferred configuration for a film thickness around 3 m23. Figure 4 (c-d) show simulated disclination line behavior of BPII (Fig. 4c) and BPI (Fig. 4d) at the proximity of the patterned surface according to their lattice orientation; for each case, the director field above the patterned surface is also shown. For certain lattice orientation the symmetry of the BP can be altered or destroyed for a given pattern parameter (Wp/Ls) while others are kept or favored by the pattern. As we previously stated, such effect of the pattern parameter over the BPlattice orientation has been captured by the free energy density difference in Fig. 4a-b. Theory and experiments indicate that the stripe pattern period, Ls, that produces the optimal conditions to nucleate and grow a BPII(100) single crystal is when Ls=aBPII with the same ratio of planar and homeotropic anchoring (Wp=WH=aBPII/2). When the pattern period varies from BP lattice constant (aBPII), other BPII-lattice orientation can be favored, as the case of BPII(110) for which it is convenient to have 𝐿𝑆~ 2𝑎𝐵𝑃𝐼𝐼 and Wp or WH ~ 2𝑎𝐵𝑃𝐼𝐼/2). Therefore, for a given BPII-lattice orientation, the pattern parameters Ls, Wp and WH, as well as the size of the crystallographic plane, are all related to the blue phase lattice constant (or to the chiral pitch). Additionally, our simulations indicate that moderate anchoring energies are sufficient to direct the 8


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nucleation of distortion free blue phase crystals. The reflection spectrums of BPI and BPII under the different pattern confinement are shown in Figure 5. BPI and BPII present highly selective light reflection because of their crystalline structure. As shown in figures, single-crystal BP domain exhibits a sharp intensity peak, being consistent with the lattice orientation over the entire pattern area. From Figure 5a, we can learn that once the system reaches to BPII temperature regime, BPII(100) is stabilized as single crystal when the pattern period is between 120 nm to 170 nm. While, the light reflective peaks are slightly different because the corresponding temperature is at the proximity of the BPI-BPII transition temperature. After a few minutes of the relaxation all the light reflection peaks highly align at 450 nm. Similar BPII spectra intensities are obtained for the same pattern periods but different planar/homeotropic stripe width ratios (Figure 5b). For the BPI, the highest intensity of light reflection is shifted to ~550 nm with a single and sharp peak, showed in green color (Figure 5c). Finally, it is appealing to draw analogies with the epitaxial assembly of block copolymers in lamellae phase, BCP, onto neutral or preferential wetting striped surfaces, where a perfect epitaxial assembly can occur up to ~±10% of mismatch between the BCP’s natural period and the underlying pre-patterns19, 20. The morphology and structure obtained from such epitaxial assembly of BCP films is determined by the relationship between the commensurability of Ls and L0, where in this context L0 is the natural period of the copolymer forming lamellar structure, the energy related to polymer chains’ stretching and compressing together with the interfacial energy, and how they influence the total free energy. In contrast, BPs consist of double twist cylinders arranged into a cubic symmetry with certain bulk period (L0). The epitaxial crystal growth of BPII(100) allows the deviation of the prepattern period (Ls) from L0 up to ~±20%, when the energetic penalty for certain lattice orientation deviating in period from L0 can be compensated by preferential interfacial energies. In addition, small free energy differences from one lattice orientation to the others would be sufficient to drive such epitaxial crystal growth of single crystals of BPs throughout the whole LC cell.

Conclusions 9



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This work demonstrates that a binary chemical pattern, although simple in geometry, can be determinant in directing the crystallographic orientation of a soft crystal at submicronic scales, and small (nanometric) variations of the pattern periodicity may completely change the crystallographic orientation of the referred crystal. Our theoretical results, serve to explain such observations in terms of the behavior of the free energy of the confined BPLC system according to the pattern parameters. We found that the role of the pattern is to set uniform conditions to promote the spontaneous and homogeneous blue-phase crystal nucleation and growth with a specific crystallographic orientation. Three transition regimes have been identified based on pattern parameters and periodicity: Ls=130-170 nm for single crystal of BPII(100); Ls=180-200 nm for single crystal of BPI(110); Ls=2L0 or 3L0 with Wp/Ls=0.5 for monodomain BPI(110). It is noteworthy to mention that the pattern dimensions can influence the crystal nucleation and growth of BPs because these soft crystals have lattice parameters of submicronic scales, this is a significant difference with respect to atomic crystals where the nucleation sites are much smaller. The process for obtaining arbitrary large BP-single crystals provided in this work allows control over the lattice orientation that we want to favor, is significantly faster than the current methods and is based on a pattern geometry that can be easily produced, these features are essential for manufacturing and modulating of optical devices and photonics.

Materials and Methods The detailed materials, sample preparation process and characterization information are available in Ref. 10. Thermodynamic description and simulation approach. In this work the confined chiral liquid crystal is modeled by means of a continuum mean field Landau-de Gennes formalism where the LC free energy, F, is described in terms of the tensor order parameter, Q, which contains the information about the structure of the phase and which is defined using the nx, ny and nz components of the director field and the scalar order parameter S as follows Qij=S (ni nj -1/3 ij). In this formalism, the free energy functional is given by24-26, 𝐹(𝑸) = ∫d3𝑥[𝑓P(𝑸) + 𝑓E(𝑸)] + ∫d2𝑥𝑓S(𝑸), 𝐴

(

𝑈

)

where 𝑓P = 2 1 ― 3 tr(𝑸2) ―

𝐴𝑈

( 3) 3 tr 𝑸 +

𝐴𝑈 4

(1) 2

tr(𝑸2) accounts for the phase contribution where A 10


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[

1

∂𝑄𝑖𝑗∂𝑄𝑖𝑗

∂𝑄𝑙𝑗

]

and U are phenomenological parameters; 𝑓E = 2 𝐿1 ∂𝑥𝑘 ∂𝑥𝑘 + 4𝐿1𝑞0𝜖𝑖𝑘𝑙𝑄𝑖𝑗 ∂𝑥𝑘 represents the elastic interactions where L1 is the elastic constant and q0=2/p; fS is the surface free energy as given in references 24-26. The minimization of the free energy is achieved by means of the GinzburgLandau method24 where the system is described using an array with mesh resolution of 7.5 nm. The anchoring energies considered in this work are EP = EH=1  10-3 J/m2 for planar and homeotropic anchoring respectively. Materials parameters are the same as in Ref. 11, i.e. A=1.067  105 J/m3, L1=6 pN, p=258 nm, U=2.755 for BPII and U=3.0 for BPI.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] , [email protected] Author Contributions X.L and J.A.M-G contributed equally to this work. X.L., J.A.M-G., J.J. de P. and P.F.N. conceived and designed the project. X.L. and P.F.N. performed the experiments. J.A.M-G, and J. J. de P. performed numerical simulations and theoretical calculations. X.L., J.A.M-G. and P.F.N. wrote the manuscript. J.J.de P. and P.F.N. guided the work. All authors discussed the results and contributed to data analysis and interpretation of the reported findings.

Notes The authors declare no competing financial interests. Acknowledgement This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Fabrication of patterned surfaces was carried out at Center for Nanoscale Materials, an Office of Science user facility, which was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, 11



under Contract No. DE-AC02-06CH11357. We acknowledge the using of computational facility for the simulation work at University of Chicago Research Computing Center. J.A.M.G. acknowledges PRODEP for providing computational resources to perform part of theoretical analysis. The SEM characterization was carried out by using MRSEC Shared User Facilities at the University of Chicago (NSF DMR-1420709). We thank Dr. David A. Czaplewski for helping with the JEOL 9300FS electron beam writer and Dr. James Dolan for useful discussions.

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(14) Wright, DC.; Mermin, ND. Crystalline liquids: the blue phases. Rev. Mod. Phys. 1989, 61, 385-433. (15) Yan, J.; Wu, ST.; Cheng, KL.; Shiu, JW. A full color reflective display using polymerstabilized blue phase liquid-crystal. Appl. Phys. Lett. 2013, 102, 081102. (16) Claus, H.; Willekens, O.; Chojnowska, O.; Drabrowski, R.; Beeckman, J.; Neyts, K. Inducing monodomain blue phase liquid crystals by long-lasting voltage application during temperature variation. Liq. Cryst. 2016, 43, 688-693. (17) Kim, K.; Hur, ST.; Kim, S.; Jo, SY.; Lee, BR.; Song, MH.; Choi, SW. A well-aligned simple cubic blue phase for a liquid crystal laser. J. Mater. Chem. C 2015, 3, 5383-5388. (18) Chen, CW.; Hou, CT.; Li, CC.; Jau, HC.; Wang, CT.; Hong, CL.; Guo, DY.; Wang, CY.; Chiang, SP.; Bunning, TJ.; Khoo, LC.; Lin, TH. Large three-dimensional photonic crystals based on monocrystalline liquid crystal blue phases. Nat. Commun. 2017, 8, 727. (19) Edwards, EW.; Montague, MF.; Solak, HH.; Hawker, CJ.; Nealey, PF. Precise Control over Molecular Dimensions of Block-Copolymer Domains Using the Interfacial Energy of Chemically nanopatterned Substrates Adv. Mater. 2004, 16, 1315−1319. (20) Liu, CC.; Ramirez-Hernandez, A.; Han, E.; Graig, GSW.; Tada, Y.; Yoshida, H.; Kang, H.; Ji, SS.; Gopalan, P.; de Pablo, JJ.; Nealey, PF. Chemical Patterns for Directed SelfAssembly of Lamellae-Forming Block Copolymers with Density Multiplication of Features. Macromolecules 2013, 46, 1415-1424. (21) Li, X.; Armas-Perez, JC.; Martinez-Gonzalez, JA.; Liu, X.; Xie, HL.; Bishop, C.; Hernandez-Ortiz, JP.; Zhang, R.; de Pablo, JJ.; Nealey, PF. Directed self-assembly of nematic liquid crystals on chemically patterned surfaces: morphological states and transitions. Soft Matter 2016, 12(41), 8595-8605. (22) Li, X.; Armas-Perez, JC.; Hernandez-Ortiz, JP.; Arges, C.; Liu, X.; Martinez-Gonzalez, JA.; Ocola, L.; Bishop, C.; Xie, HL.; de Pablo, JJ.; Nealey, PF Directed self-assembly of colloidal particles onto nematic liquid crystalline defects engineered by chemically patterned surfaces. ACS Nano 2017, 11 (6), 6492–6501. (23) Bukusoqlu, E.; Martinez-Gonzalez, JA.; Wang, XG.; Zhou, Y.; de Pablo, JJ.; Abbott, N. Strain induced alignment and phase behavior of blue-phase liquid crystals confined to thin films. Soft Matter 2017, 13, 8999-9006. (24) Ravnik, M.; Žumer, S. Landau-de Gennes modeling of nematic liquid crystal colloids. Liq. Cryst. 2009, 36, 1201-1214. (25) Alexander, G. P.; Yeomans, J. M. Numerical results for blue phases. Liq. Cryst. 2009, 36, 1215-1227. (26) Dupis, A.; Marenduzzo, D.; Yeomans, J. M. Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2005, 71, 11703. (27) Fournier, G.; Galatola, P. Modeling planar degenerate wetting and anchoring in nematic liquid crystals. Europhys. Lett. 2005, 72, 403-409.

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Figures and captions:

Fig. 1 a) Schematic representation of the fabrication process to create chemical patterns with regions of competing anchoring properties from PMMAZO brush substrates having nano- or micro-scale straight stripes. The homeotropic anchoring glass substrate was placed face-to-face 14


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with the patterned substrate to form a hybrid cell and LCs were injected into the gap through capillary action. b) Blue phase materials, MLC2142 mixing with chiral dopant S-811, formed cholesteric phase, BPI, BPII and isotropic by increasing the sample temperature. The stripe-like patterns with different pattern dimensions are used to form a stable single-crystalline BPI(110) at 40.1 oC and BPII(100) at 41.7 oC. The corresponding Kossel diagrams are included to verify the lattice symmetry.

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Fig. 2 a) Schematic representation of the chemical patterned surface showing the pattern condition of Ls=L0=150 nm. b) Free energy density as a function of Wp/Ls for a pattern period of Ls=150 nm (BPII), showing that the patterned substrate favors the (100)-lattice orientation. c) Reflective light microscope images of the Blue phase cell under crossed polarizers, with stripe pattern period as 16


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150 nm, and Wp/Ls varies from 0-1. d) BPII-disclination lines (visualized as nematic order parameter isosurfaces with S=0.35) and the preferred director alignment at the interface for different Wp/Ls ratios and lattice orientations. Color map goes from blue (0) to red (1.0) and is associated to the projection of the local director to the z-axis, therefore, blue corresponds to the case where the alignment is parallel to the surface while red corresponds to the perpendicular case.

Fig. 3 a) Images of the Blue phase cell taken in reflection mode under crossed polarizer microscope, with Wp/Ls=0.5 and stripe pattern period changing from 130 nm to 450 nm. b) Reflective light microscope images of the Blue phase cell under crossed polarizer, with Wp=L0/2 and stripe pattern period changing from 130 nm to 450 nm, and the width of planar anchoring stripe is kept as 75 nm (L0/2). Kossel diagrams of BP phase of corresponding BPs are shown at the left corner in the images.

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Fig. 4 a-b) Free energy density as a function of Wp/Ls for a pattern period of Ls=200 nm (BPII) and Ls=180 nm (BPI). As can be seen from the figure, for 0.4

Perfection in Nucleation and Growth of Blue-Phase Single Crystals

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