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Smectic Gardening on Curved Landscapes Mohamed Amine Gharbi, Iris Bi Liu, Yimin Luo, francesca serra, Nathan D Bade, Hye-Na Kim, Yu Xia, Shu Yang, Randall D. Kamien, and Kathleen Joan Stebe Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b02508 • Publication Date (Web): 23 Sep 2015 Downloaded from http://pubs.acs.org on September 27, 2015

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Smectic Gardening on Curved Landscapes Mohamed A. Gharbi,

Bade,

‡

Hye-Na Kim,

§

†

Iris B. Liu,

Yu Xia,

§

‡

Yimin Luo,

‡

Francesca Serra,

Randall D. Kamien,

Stebe

¶

Shu Yang,

§

‡, ¶ , §

Nathan D.

and Kathleen J.

∗,‡

Department of Physics, McGill University, Montr e´al, Que´bec, Canada, Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, and Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia E-mail: [email protected] Phone: 1-215-898-4515. Fax: 1-215-573-2093

Abstract Focal conic domains (FCDs) form in smectic-A liquid crystal lms with hybrid anchoring conditions with eccentricity and size distribution that depend strongly on interface curvature. Assemblies of FCDs can be exploited in settings ranging from optics to materials assembly. Here, using micropost arrays with dierent shapes and arrangement, we assemble arrays of smectic ower patterns, revealing their internal structure, as well as defect size, location, and distribution as a function of interface curvature, by imposing positive, negative, or zero Gaussian curvature at the free surface. We characterize these structures, relating free surface topography, substrate anchoring ∗ † ‡ ¶ §

To whom correspondence should be addressed

e´

e´

Department of Physics, McGill University, Montr al, Qu bec, Canada Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia Department of Physics and Astronomy, University of Pennsylvania, Philadelphia Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia

1

ACS Paragon Plus Environment

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strength and FCD distribution. While the largest FCDs are located in the thickest regions of the lms, the distribution of sizes is not trivially related to height, due to Apollonian tiling. Finally, we mold FCDs around microposts of complex shape, and nd that FCD arrangements are perturbed near the posts, but are qualitatively similar far from the posts where the details of the conning walls and associated curvature elds decay. This ability to mold FCDs defects into a variety of hierarchical assemblies by manipulating the interface curvature paves the way to create new optical devices, such as compound eyes, via a directed assembly scheme.

Introduction There is intense interest in using liquid crystals (LCs) and their associated director elds and defect structures as recongurable soft matter. The optical and elastic properties associated with LCs are exploited in a wide variety of settings ranging from optical applications to directed assembly of colloids and nanoparticles.

14

Among the geometric defects, focal

conic domains (FCDs) are particularly interesting in this context. FCD, called type I FCD conditions.

57

5

One particular type of

forms in smectic-A (SmA) LC lms with antagonistic boundary

FCDs consist of molecular layers wrapping around two curves, an ellipse

and a hyperbola, which contain all singular points (see inset in

Figure 1(a)); these Type 1

defects deform the air-LC interface and decorate the surface with dimples.

8,9

Under appro-

priate conditions, FCDs assemble into arrays whose features have been exploited in several settings. For example, colloids become trapped in individual defects,

2

driven by the exquisite

sensitivity of the elastic energy eld to perturbation. The arrays are used as precursors of super-hydrophobic surfaces

10

and as templates for soft lithography

11

owing to the periodic,

micro-scale textures of their free surfaces. In addition, FCDs are exploited in photonic and lasing applications

12

and as elements of bistable displays.

13,14

Individual FCDs within an

array can serve as microlenses for imaging or as photomasks, enabled by the modulation of the refractive index within the layered structure of the defect.

2


1517

These applications rely

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on FCD size, which depends on the lm thickness, which can be modulated by connement,

15,16

and on their spatial arrangement,

e.g., in channels or thin lms.

Important new degrees of freedom emerge when we depart from at interfaces. For example, on a sessile drop or when a curved lm of liquid forms around a colloid placed in a thin smectic lm, a ower pattern of defects emerges.

This pattern has elegant symmetries

and a hierarchical distribution of FCDs of diering sizes and heights, with individual defects resembling ower petals.

8

Beller

et al.

constructed a theoretical model for this pattern in

which the location of singularities arranged around the conic sets of ellipses and hyperbolae are constrained so that smectic layers are continuous and equally spaced, and such that the ellipses and hyperbolæ of the FCDs satisfy the law of corresponding cones.

18

If, instead of

an isolated colloid, smectic lms are formed around microposts arranged in regular arrays, ower patterns emerge around each micropost (Figure 1(a)); in this setting, the interface curvature, and hence the FCD spatial arrangement in each ower can be better controlled. By exploiting a uorinated liquid crystal which maintains the smectic phase in a glassy state at room temperature, we have been able to directly visualize the structure of the smectic layers in the ower pattern around a micropost by Scanning Electron Microscopy (SEM) (Figure 1(b)-(c); See Experimental section for details).

9

The FCD ellipse lays on the bottom

substrate, with planar anchoring conditions, and the defect core forms a hyperbola pointing away from the micropost, as predicted by the theoretical model. Such assemblies have unique optical properties determined by the size of the defects and the curvature of the lm. The bending of the layered structure inside each individual FCD modulates the local index of refraction, a property that we have exploited by using the ower pattern as an assembly of microlenses that resembles the compound eyes of invertebrates (Figure 1(d)-(e)).

Image plane heights, size distributions, and focal lengths are all

tunable parameters of these microlenses.

19

The lenses are also thermally recongurable and

sensitive to light polarization. Furthermore, FCDs in the ower pattern can trap colloids at their core, allowing nontrivial colloidal assemblies (Figure 1(f )).

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Figure 1:

Structure and applications of FCDs.

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a) Schematic of an individual focal

conic domain and the arrangement of FCDs along a curved interface around a micropost. b) Cross-sectional SEM image of the uorinated smectic LC lm around a micropost. The micropost wall is on the right (not depicted). It is possible to distinguish the ellipses (solid red lines) and hyperbolae (dashed red lines) of the FCDs, the latter pointing outwards. The two left most FCDs are well resolved after fracturing the sample. They show the decrease in FCD size with interface height. The right most FCD is less well resolved. c) SEM image of uorinated smectic LC around a micropost under SEM. d) Schematic of the microlens experiment setup. e-f ) Bright-eld optical microscopy images of e) FCDs around a micropost that act as a microlens array, imaging the Ps above the FCDs, and f ) 2 are trapped in FCDs, as pointed by the arrows. The scale bars are 10 in c), e), and f ).

4


µm

µm

silica particles

in b) and 50

µm

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In these motivating examples, the highly-regular FCD assembly results from the interplay of interface curvature, geometry of the conning structure, lm thickness, elasticity, and surface anchoring strength, among other factors.

The role of such factors in dening

FCD structure is largely unexplored. In this paper, we create a garden of liquid crystal owers by conning a smectic lm on a substrate templated with micropost arrays and other topographical cues prepared by lithography, which are treated to impose dierent anchoring strengths. The cues are designed to generate air-LC interfaces with zero, positive, and negative Gaussian curvatures, dened as the product of the principal curvatures. The sign of this quantity is known to play a strong role in the arrangement of related systems including the packing of foams,

20,21

the arrangement of conned colloids,

of block co-polymers on curved substrates.

24

22,23

and arrangements

Our approach suggests a facile way of creating

hierarchical assemblies of microstructures by engineering the shape of the interface.

Experimental Sample preparation We rst fabricated an array of microposts on a glass substrate using standard lithographic techniques

19

with negative tone epoxy, either KMPR or SU-8 (MicroChem Corp).

microposts, arranged in a square array, have height h and pitch p

≈

300

µm.

≈

40

µm,

diameter D

≈

100

The

µm

A layer of smectic LC, 4-cyano 4'-octylbiphenyl (8CB, Kingston

Chemistry, Inc) in the isotropic phase is introduced into the space between the microposts by casting the SmA LC and leveling with a coverslip; in this way the smectic lm pins at the top edge of the microposts. The system is maintained at approximately

TSmA−N = 33.4◦ C, 25

the nematic to smectic transition temperature, for about 30 minutes, and then allowed to cool gradually to room temperature to form the SmA phase. FCDs form in the lm around the microposts.

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Anchoring conditions To form FCDs, the lm must have competing boundary conditions,

i.e.,

homeotropic at

the air-smectic interface and degenerate planar in contact with the microposts and glass substrate. strength.

Three dierent surface treatments are used to vary the substrate anchoring In the rst substrate preparation, microposts are made with KMPR without

subsequent modication; this is considered the standard condition.

In the second prepa-

ration, we sputter the KMPR posts with chrome and subsequently drop-cast polyvinyl alcohol (PVA). In the third preparation, we treat the KMPR microposts with APTES, (3-Aminopropyl)triethoxysilane (Sigma-Aldrich).

These latter preparations induce degen-

erate anchoring of dierent strengths. We estimate the anchoring strength by the method suggested by Guo and Bahr:

26

r=( where

r

is the radius of FCDs and

6∆σ 1 )2 h σair

h is the lm thickness.

anchoring strength at the free smectic surface and the substrate with

∆σ = σsub−Sm − σair .

(1)

In this expression,

σsub−Sm

σair

denotes the

denotes the anchoring strength on

While this relation has been explored only for at

lms, and our system has a curved interface, we use this argument for guidance, where we consider

h to be the interface prole that, in turn, inuences the local FCD radius.

Since the

interface height prole is determined by pinning the smectic lm to a micropost of a given height, we assume it is similar for all anchoring conditions.

Furthermore, the air-smectic

anchoring strength is assumed to be the same in all cases (about 30 mN/m implies that larger FCDs are a result of stronger anchoring at the substrate.

2628

).

This

Using FCD

radii observed around the microposts with diering surface treatments, we infer estimates for the anchoring strengths are

σP V A−Sm

= 1.4 mN/m and

σAP T ES−Sm

= 1 mN/m. We note

here that PVA-treated substrates behave similarly to those prepared with untreated KMPR. When measuring the size distribution, we use an image processing software, ImageJ, to calculate the average size of FCDs located at a given distance from the post. In this case we

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consider all the FCDs, which is discussed later in detail. In order to estimate the anchoring strength, however, we only rely on the largest FCDs, as the small FCDs which form the Apollonian tiling do not penetrate all the way in the sample and, therefore, they do not feel the anchoring strength of the planar substrate.

Fluorinated molecule for scanning electron microscopy To visualize the section of the smectic LC layer around pillars, semi-uorinated smectic liquid crystal, (4'- (5,5,6,6,7,7,8,8,9,9,10,10,11,-11,12,12,12-heptadecauorododecyloxy)biphenyl-4-carboxylic acid ethyl ester), was employed to form FCDs; this LC was heated to its isotropic state (200

◦

C) by 5

◦

◦

C) and cooled down to the smectic transition temperature (114.0

C/min. The molecules were synthesized by a two-step reaction and purication

following the literature.

2,9,29,30

The samples were broken manually and residual debris was

removed with tape. They were then characterized by SEM.

Characterization techniques Initial characterization of the FCDs was performed using an upright microscope (Zeiss AxioImager M1m, Carl Zeiss Microscopy LLC., Thornwood, NY) in transmission mode equipped with crossed polarizers. The images were captured with a high-resolution camera (Zeiss AxioCam HRc). Objective magnication ranged from 10x to 100x. The surface prole of the ower pattern formed by FCDs was obtained by a Zygo NewView 7300 3D optical proler (Zygo Corporation, Middleeld, CT), using 20x and 50x Mirau objectives.

SEM

images were acquired with a Quanta 600 FEG Mark II microscope equipped with a low voltage detector for environmental SEM (Singh Nanoscale Characterization Facility at the University of Pennsylvania). We used 10 kV voltage and about 1 Torr of chamber pressure, and the sample was not coated with any metal layer.

Confocal microscopy was used to

obtain the three-dimensional reconstruction of the owers. The system was prepared on a glass slide and loaded with LC molecules doped with 0.01 wt % of the uorescent dye N,N'-

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bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylene-dicarboximide (BTBP, Sigma Aldrich).

The

ower texture was visualized with a Leica TCS SP8 confocal microscope (Leica Microsystems, Bualo Grove, IL; Cell & Development Biology Microscope Core at the University of Pennsylvania) using an excitation laser line at 488 nm. The scans presented in this paper were performed using either a 20x air objective or a 25x water immersion objective.

To

minimize noise, 2x the line average and 2x the frame average are performed during the scan. These resolutions varied from image to image according to the objective. The image stacks were captured by the Leica Acquisition software (LAS) and post-processed with Fiji software. Image J-based open-source software was used for quantitative analysis. The data from all four dierent scans are collapsed into one plot, where micropost, and

∆z

r

is the distance from center of the

is the height of the interface minus the baseline. A custom MATLAB

code was written to analyze the results from the maximum intensity confocal stack. details of prole analysis and the code can be found in

The

Supplementary Information

1

and 2.

Results and discussion Focal Conic Domain Characterization Interfaces around an individual micropost have negative Gaussian curvatures, as do locations where lms from two adjacent microposts merge. Sessile drops form interfaces with positive Gaussian curvature. Finally, isotropic wetting lms near at walls have zero Gaussian curvature. The "envelope interfaces of smectic lms have proles similar to those of isotropic uids, but are punctuated with topographies and dimples associated with the FCDs. The interface has some dependence on anchoring properties that has not been explored to our knowledge, but it essentially behaves as an isotropic uid interface within our range of lm thicknesses. With this as a preamble, we describe observed FCD arrangements in each of the curvature zones. The case of very thin smectic lms is beyond the scope of this paper

8


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and is discussed in detail elsewhere.

31

We focus rst on smectics conned in micropost arrays. Below a minimum lm thickness, FCDs do not form: the molecules break free of the planar anchoring on the substrate and form uniform smectic layers. When the minimum thickness is high enough to support FCD formation (typically, 2-3 micropost (

µm 26 ), FCDs appear and form isolated ower patterns around each

Figure 2(a) and (b)).

By increasing the LC volume, adjacent menisci thicken

and merge, allowing a continuous garden of connected owers to form (Figure 2(c) and (d)). Gradients in lm height allow both isolated owers and connected owers to occur in the same sample (Figure 2(e)). In the bowl-like region between four microposts, the lm is thinnest at the center and thickens near the edges. In lms with this geometry that are suciently thick, FCDs arrange with their smallest members in the center and their largest toward the edges. Sessile drops atop microposts form owers with the largest FCDs toward the center of the drop and the smallest near the edge (Figure 2(c) and (d)). Finally, the edge of a at wall can also act as a pinning site, inducing a hierarchical arrangement of FCDs (Figure 2(h)).

In all of these assemblies, the interstices between large FCDs are lled by smaller

FCDs as in the Apollonian tiling of the so-called generation pattern

32

(clearly visible in

Figure 2(f )). While FCD formation is driven by the hybrid texture of the smectic lm, both the interface curvature and lm height dictate the defect size, packing, and assembly. The FCDs are largest near the micropost, where the lm is thickest. The slope is also highest in this region; consequently, the FCDs formed there have more eccentric ellipses. To obtain detailed information about the 3D structure of the owers, confocal microscopy, optical interferometry, and atomic force microscopy are used to locate the free surface.

ure 3(a)

Fig-

shows the height prole of four isolated owers and a red line that is intended

to guide the eye.

Despite the variability across the samples due to small changes in the

volume of SmA LC spread on the microposts, the data collapse onto a single prole shape starting at a distance of around 10

µm

from the micropost edge. Confocal microscopy data

are used to generate a 3D reconstruction of a connected ower with a sessile drop on top

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Figure 2:

tures.

Page 10 of 25

Flower textures form in regions around microposts with various curva-

The interface shape is controlled by the volume of SmA LC in the cell. a) Bright

eld and b) polarized optical microscopy images of an isolated smectic ower. c) Bright eld and d) polarized optical microscopy images of nine connected owers in a micropost array. e) A row of isolated owers next to a row of undulated owers with connected menisci. f ) Bright eld and g) polarized microscopy images of the ower texture in a sessile drop on top of a micropost with degenerate planar anchoring conditions. h) Polarized optical microscopy image of FCDs near a at wall. The scale bars are 100

µm

a), b), f ), g), and h).

10


in c), d), and e) and 50

µm

in

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of the micropost (Figure 3(b)). Owing to the complexity to this image, we describe it here in some detail. In the top image of Figure 3(b), the center of the relatively dark circle is the top of the pillar which is partially covered by a sessile drop, which appears as bright uorescent domains surrounded by a nearly black ring.

Outside of this ring, a gray zone

appears; this is the top of the free surface with negative-Gaussian curvature surrounding the pillar. In this region, the slope of the interface is steepest and artifacts from scattering are most signicant. At greater distances from the center of this image, the FCDs distributed on the curved interface are evident. In the bottom image, corresponding side views of the bright sessile sessile drop, and high scattering regions that are free of uorescent signal are evident. These are above the bright signal from the FCDs distributed along the negative Gaussian curvature interface. Interferometry data indicates that the envelope interface shape is more complex around connected owers exhibiting a four-fold symmetry around a central post (Figure 3(c)). The region between two adjacent microposts has a saddle point as shown in

Figure S1.

The sessile drop has a small, but clearly positive curvature as shown in the

interferogram in Figure 3(d). The structure of the owers is driven not only by capillarity, but also by elastic forces that depend weakly on the strength and type of anchoring on the substrate. In order to quantify this eect, we treated substrates with polyvinyl alcohol (PVA) and (3-aminopropyl)triethoxysilane (APTES) to impose strong and weak planar anchoring, respectively. As a control, we also tested microposts with homeotropic anchoring, which did not form FCDs. The sample with weaker anchoring (

Figure S2) has a slightly slower decay of the envelope interface prole

with distance from the micropost.

It forms qualitatively similar ower structures (Figure

S3), both around posts and in sessile drops, albeit with smaller and more numerous FCDs in each ower. Homeotropic anchoring, which, as expected, does not induce the formation of FCDs, induces a slower decay of the interface prole with distance (

Figure S3).

An interesting observation emerges when we look into the detailed structure of the interface owing to the presence of the FCDs. In prior work on FCD arrays, FCD surfaces have

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Figure 3:

Characterization of 3D ower structures .

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a) The envelope interface shape.

Blue, green, turquoise, and black dots represent height information obtained from collapsing four confocal stacks and deducing an average prole shape of the meniscus around the micropost. The red line is a guide for the eye. b) Reconstructions of the ower from an oblique angle and cross-sectional view from confocal microscopy. c) Interferogram showing the shape of the SmA interface for a connected ower with negative curvature. d) Interferogram of the interface shape of a sessile drop on top of a post. e) Atomic force microscopy (AFM) image of an individual FCD on the curved interface. f ) Height prole of the AFM scan across the center of the FCD. Inset: An interferogram of the FCD showing the typical shape is similar to the AFM prole. The scale bars are 50

µm

in (b-d) and 5 µm in (e).

12


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been reported to have a convex dimple at the hyperbolic defect core, as shown in Figure 1(c) from the uorinated SmA LC. Surprisingly, our interferometric images and supporting AFM data (Figure 3(e)) on 8CB reveal that individual FCDs have a concave free surface, as appears from the AFM data in Figure 3(f ) which terminates in a convex dimple at its core. The boundary between each FCD appears to be well dened and smooth, such that the overall shape of the interface of a sessile drop resembles the surface of a golf ball (Figure 3(d)). This FCD concave shape was observed both on the sessile drop and around the microposts. A careful review of the literature reveals that this unexpected shape of the FCDs was observed in previous studies

7,33,34

on very thin lms of smectics (much thinner than those

employed in this work), but it was never fully explained.

A deeper understanding of this

phenomenon and its implications are currently under investigation, but this topic is beyond the scope of this paper.

Packing Characterization In the above discussion, we have shown that the shape of the air-LC envelope interface has a profound impact on the arrangement of FCDs within an LC lm. Here, we further investigate the size distribution and the packing of FCDs in four scenarios: (i) an isolated ower, (ii) a connected ower, (iii) a sessile drop on top of a micropost, and (iv) in a meniscus along a wall. The envelope interface shape has negative Gaussian curvature in (i) and (ii), positive Gaussian curvature in (iii), and zero Gaussian curvature in (iv). extracted and analyzed the fraction

φ

of area occupied by FCDs as a function of the radial

distance from the center of the micropost ( regions of high

From confocal data, we

Figure 4).

FCDs which pack eciently form

φ; below, we show that the packing is not trivially related to interface height.

Around isolated microposts (Figure 4(a)), the largest FCDs form closest to the micropost where the lm is thickest, while far from the micropost the majority of the FCDs are small (< 15

µm2 ).

However, the packing and distribution of FCDs is not trivially related to the lm

height owing to tiling eects. The region close to the microposts contains a few very large

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Figure 4:

Size distribution analysis of owers with dierent curvatures.

Page 14 of 25

a) Isolated

posts. b) Connected posts. c) Spherical caps. For each shape, we have shown in column 1: our numerical code sets the concentric annuli for distance determination, superposed on the confocal microscopy image of the ower; column 2: bar chart showing normalized count of objects with respect to radial distance, categorized by FCD area; column 3: percentage areas occupied by FCDs with respect to distance from the features.

14


average

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FCDs, but also many small FCDs that help the packing as in an Apollonian tiling.

µm2 ).

near the microposts, the majority of FCDs are also small (< 15 region contains a larger fraction of medium (15-40

32

Thus,

The intermediate

µm2 ) and large (> 40 µm2 ) FCDs.

Because

of these packing eects, the fraction of area occupied by FCDs

φ

with distance from the post and hence with interface height.

It is greatest at a distance

of approximately 85

µm

from the center of the post (

i.e.

35

µm

varies non-monotonically

from its edge), within the

intermediate region containing large FCDs. In our previous work on the optical properties of FCDs, we observed that this region also yields the best lensing eect in terms of image resolution.

19

FCDs are arranged similarly in connected owers, except that the size distribution is broadened because the interface decays less with distance (Figure 4(b)). position of the most eciently packed region shifts from about 85 the posts to approximately 110

µm.

µm

Moreover, the

from the center of

The size proves to be sensitive to slight dierences in

the volume of LC, as are both the interface shape and the spatial arrangement of FCDs. Again,

φ

is greatest in the region containing large FCDs, but is only half of that observed

for the isolated owers (i.e. the maximum value for

φ

is

∼

0.3 in connected owers vs. 0.6

in isolated owers). The large dierence could be attributed to the high slope of the isolated owers favoring the formation of smaller FCDs that pack more eciently. FCDs conned beneath positive Gaussian curvature interfaces of sessile drops organize very dierently. The largest FCDs are at the center of the droplet and the fraction of small FCDs increases monotonically with radial distance from the center (Figure 4(c)). The FCDs pack eectively to yield highest

φ

at the drop center, where the drop is tallest;

φ

decreases

monotonically with drop height from this region. FCDs near a at wall (

Figure 5(a)) show a trend in packing eciency similar to that of

isolated microposts, with a characteristic region of ecient packing located about 100 away from the wall (Figure 5(b)).

µm

Close examination of a scatter plot of FCD radii as a

function of distance from the wall reveals two distinct trends (Figure 5(c)). First, the sizes

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Figure 5:

Page 16 of 25

Size distribution and packing analysis of FCDs near a at wall.

a) FCDs

near a at wall. The red circles indicate the objects captured from a confocal microscopy stack by the custom code. The scale bar is 50

µm.

b) Area occupied by FCDs as a function

of the distance from the wall. c) FCD radii as a function of distance from the wall. Each point in the plot corresponds to one FCD.

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of the largest FCDs decrease almost linearly with distance from the wall, as expected.

34

Second, it is apparent that a distinct population of small FCDs are involved in Apollonian tiling.

The size distribution of FCDs within this region has a parabolic shape, with the

maximum of the curve coinciding with the maximum area fraction occupancy by FCDs as shown in Figure 5(b). To summarize, it appears that the packing eciency of FCDs is crucially dependent on the size of the FCDs that ll the gaps between larger domains. While Apollonian tiling does not aect the general height/width relationship of the larger FCD, it is essential for the optimal packing in the ower pattern. It is worth noting that the owers were all formed in the same experimental conditions; in particular, the rate of cooling of the LC into the SmA phase was consistent throughout the experiments. The cooling rate aects the size distribution: a slower cooling rate allows for the formation of larger FCDs and simultaneously suppresses the formation of small FCDs.

Complex-shaped Flowers The shape of bounding surfaces inuences the arrangement of the SmA LC. In particular, geometric singularities, such as corners and edges, can induce unexpected behaviors in related systems.

6

To understand the role of these features in our system, we study the formation

of owers around microposts with vertices and edges.

To do so, we fabricate posts with

triangular, square, and pentagonal cross sections. We also fabricate star-shaped microposts with 3, 4 and 5-arms. This variety of shapes allows us to analyze the eect of convex and concave angles

α

dened at the corners where edges meet, as shown in

Figure 6.

A few key observations can be gleaned from the images in Figure 6. First, we discuss microposts with convex edges ( α

< 180◦ , as in Figure 6 (a),(c), and (e)).

Interface curvature

varies steeply near corners, therefore FCD size decays rapidly with distance from corners and closely follows the contour of the posts (

Figure S4).

However, over a distance of

only a few FCD widths from the post, this mimicry is lost (especially in the pentagonal micropost, as shown in (

Figure S5),

and FCDs form in packings similar to those around

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circular microposts.

Page 18 of 25

This observation has analogy to prior work on capillary interactions

around square microposts. Interface curvature elds owing to sharp corners decay rapidly with distance from the post; beyond a certain distance, the curvature elds and associated capillary interactions are essentially the same as those around circular microposts. Here, the eect of sharp corners on the arrangement of FCDs is also highly local. In the microposts with concave angles ( α

> 180◦ ,

as in Figure 6 (b),(d), and (f )), the

previous observation still holds: the FCDs tend to heal the corners.

However, another

behavior is evident: the concave angles are lled by very large and distorted defects. The ensemble of the topographic features arrange in a way that large FCDs ll in the interior corner in order to recover the corresponding shape with convex angle ( plus the large internal FCDs resembles the triangular post,

etc.).

e.g.

, the 3-arm star

In this way, the distribution

of the outer FCDs is almost unperturbed by the concave regions. With this small exploration across dierent topographic templates, we can observe that the ower pattern, intended as a hierarchical assembly of FCDs with dierent eccentricity on a curved surface, is quite ubiquitous and it is not easily disrupted by the vertices and edges of the microposts. On the contrary, the assembly of the FCDs always tends to recover a circular arrangement, independent of the shape details of the topographic features.

Conclusions We investigated topological cues to ensembles of FCDs by imposing anchoring conditions at the substrate and molding the interface shape on micropost arrays with positive, zero, and negative curvatures (all simultaneously present if the smectic lm is formed on top and surrounding a micropost array) with dierent surface treatment and complex shapes. When hybrid anchoring conditions are imposed at the boundaries, the curved interfaces we create lead to the formation of ower structures and the envelope interface shape does not deviate from that of an isotropic uid signicantly. However, we found that in the case of

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Flower texture formed around other geometries seen in bright eld microscopy. In all cases the inclusion has degenerate planar anchoring. In the near Figure 6:

eld, the inclusion causes the focal conic domains to arrange in distinct ways. However, in the far eld, the smectic lm rounds the corners so that the distribution of the FCDs is axially symmetric.

The cross sections of the microposts are a) triangular, c) square, and

e) pentagonal. Flower textures also form around more complicated geometries with various numbers of interior corners where the cross sections of the pillars are b) three-armed, d) four-armed, and f ) ve-armed stars. The microposts have lateral size of about 100 height of 40 bars are 50

µm, comparable in size to the cylindrical microposts discussed above. µm.

19


µm

and

The scale

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Page 20 of 25

8CB owers, the local deformation of the FCDs surface took an unexpected concave shape. The spatial distribution and the packing fraction of FCDs are very dierent for negative vs. positive Gaussian curvatures.

Our data suggested that the packing eciency was strictly

related to the ability of smaller FCDs to form in the interstices between larger FCDs with an Apollonian tiling. Finally, we showed that the ower structures were indeed robust in the presence of corners and edges in the microposts, and they tended to heal the corners and retrieve a radial distribution. The ability to direct the smectic ower formation by topographical features demonstrated here oers new insights to manipulate the packing of FCDs in complex geometries and investigate the preservation of the continuity of the smectic layers. In turn, we expect to develop new applications from smectic owers,

e.g.,

compound eye-like microlens arrays

19

and complex assemblies of colloids and nanoparticles.

Acknowledgement M.A. Gharbi, I.B. Liu, and Y. Luo contributed equally to this work. The authors thank Nader Engheta for stimulating discussions, Andrea Stout and Jasmine Zhao for help with confocal microscopy, and Robert Carpick for access to his interferometer.

I.B.L. and N.D.B. are supported by a Department of Education GAANN Grant

P200A120246. This work was supported by the National Science Foundation (NSF) Materials Science and Engineering Center (MRSEC) Grant to University of Pennsylvania, NSF1120901. This work was partially supported by a Simons Investigator grant from the Simons Foundation to R.D.K.

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Smectic Gardening on Curved Landscapes - ACS Publications

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