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Sharp morphological transitions from nanoscale mixedanchoring patterns in confined nematic liquid crystals Julio C Armas-Pérez, Xiao Li, José A. Martínez-González, Coleman Smith, Juan P. Hernandez-Ortiz, Paul F Nealey, and Juan J. de Pablo Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02522 • Publication Date (Web): 25 Sep 2017 Downloaded from http://pubs.acs.org on September 26, 2017

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Sharp morphological transitions from nanoscale mixed–anchoring patterns in confined nematic liquid crystals Julio C. Armas-Pérez,† Xiao Li,† José A. Martínez-González,† Coleman Smith,† J. P. Hernández-Ortiz,‡ Paul F. Nealey,∗,† and Juan J. de Pablo∗,† †Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA ‡Departamento de Materiales, Universidad Nacional de Colombia, Sede Medellín, Medellín, Colombia ¶Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA E-mail: [email protected]; [email protected] Abstract Liquid crystals are known to be particularly sensitive to orientational cues provided at surfaces or interfaces. In this work, we explore theoretically, computationally and experimentally, the behavior of liquid crystals on isolated nanoscale patterns with controlled anchoring characteristics at small length scales. The orientation of the liquid crystal is controlled through the use of chemically patterned polymer brushes that are tethered to a surface. This system, can be engineered with remarkable precision and, the central question addressed here, is whether a characteristic length scale exists at which information encoded on a surface is no longer registered by a liquid crystal. To do so, we adopt a tensorial description of the free energy of the hybrid liquid crystal-surface system, and we investigate its morphology in a systematic manner. For long and narrow surface stripes, it is found that the liquid crystal follows the

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instructions provided by the pattern down to 100 nm widths. This is accomplished through the creation of line defects that travel along the sides of the stripes. We show that a “sharp” morphological transition occurs from a uniform undistorted alignment to a dual uniform/splay-bend morphology. The theoretical and numerical predictions advanced here are confirmed by experimental observations. Our combined analysis suggests that nanoscale patterns can be used to manipulate the orientation of liquid crystals at a fraction of the energetic cost that is involved in traditional liquid crystalbased devices. The insights presented in this work have the potential to provide a new fabrication platform to assemble low power bi-stable devices, which could be reconfigured upon application of small external fields.

1

Introduction

Past work has documented how the chemistry and topography of substrates can be altered to control the orientation of liquid crystal (LC) films. Most literature studies have relied on micro-scale topological features to induce LC alignment, but several researchers have also considered nanoscale patterns. Early studies resorted to polymeric surfaces rubbed along specific directions to create microscopic grooves 1,2 , and to metal vapor-deposition to create random patches of perpendicular anchoring on a planar substrate 3 . More recent efforts have relied on soft lithography 4–9 and photo-alignment 10 techniques to create chemical or topographic patterns that induce different types of anchoring on surfaces. Patterned substrates are known to induce formation of what are often referred to as multistable “phases”; examples include the tristable device by Kim et al. 5 , the fiber-induced bistable cell by Mizoshita et al. 11 or the zenithal bistable device (ZBD) of Spencer et al. and Parry-Jones et al. 12,13 . The central question that is addressed here is whether precisely patterned nanoscale isolated features with dimensions approaching characteristic molecular length scales, where anchoring is controlled through the chemical composition of polymer brushes, can be used to control LC orientation, and whether they can be used to form mixed morphologies that could offer advantages for the design of optical devices. 2 ACS Paragon Plus Environment

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The equilibrium alignment of a liquid crystal is controlled by a balance between entropic (elastic), enthalpic, and surface interactions. From a purely elastic point of view, a nematic liquid crystal’s resistance to mechanical deformations is quantified through an elastic coefficient denoted by K (representing the splay, bend and twist modes), which for thermotropic liquid crystals is of order 10−11 N. A characteristic enthalpy, denoted by A, measures the energy per volume required to melt a liquid crystal. It is of order 105 J/m3 . A p natural and intrinsic scale is the nematic correlation length, given by ξN = K/A ≈ 10 nm, can therefore be used to measure the characteristic dimensions of a liquid crystal defect (defect size is ≈ 3 nm). Past experimental work places ξN in the range from 10 to 50 nm 14,15 . Once LC molecules are in contact with a surface, they follow a specific orientation that is referred to as the “easy axis". Elastic deformations may induce deviations from the “easy axis" in the areas adjacent to the substrate. The resulting surface energy is quantified by the so-called anchoring strength, W . Experimental measurements place W in the range from 10−6 J/m2 for “weak” anchoring, to 10−2 J/m2 for “strong” anchoring. Consequently, an additional length scale arises, the so-called surface extrapolation length, given by ξS = K/W . It ranges between 1 nm for strong anchoring to 1 µ m for weak anchoring, and measures the distance from the surface where the LC direction would coincide with the “easy axis" 16 . In particular, if W → ∞ then ξS → 0 and the LC orientation is along “easy axis" at the substrate (infinite anchoring conditions). The focus of this work is to characterize the morphological transitions that a nematic liquid crystal undergoes when confined in a hybrid slit cell of thickness H, where isolated and well-defined stripes of planar anchoring and width SW are placed in a background of homeotropic anchoring. Building on past studies of LCs on nano-scaled stripes 5,6,17–23 , we consider primarily surface features and slits whose length scale (SW and H) is of the same order of magnitude as ξN and ξS . In a departure from past work, however, we examine the behavior of LCs on stripe-patterned polymer brushes, which have received considerable attention for directed assembly of block copolymers in commercial devices 24–26 , and which


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offer promise for development of new optical technologies. Simple mechanical arguments suggest that if SW ≪ ξN , ξS , the LC will ignore the anchoring and the morphology will be dominated by elastic forces that promote a uniform undistorted alignment. Conversely, if SW ≫ ξN , ξS , a mixed morphology, consisting of regions of uniform and non-uniform alignments, will be induced due to the LC propensity to follow the mixed orientations at the surface. In the geometry we are considering, the non-uniform alignment is the socalled splay-bend morphology because LC molecules, in order to satisfy the planar and homeotropic orientations at the surfaces, they undergo a splay-bend deformation. The ratio between the confinement H and the surface extrapolation length at every surface will also influence the type of LC morphology. For instance, as H approaches ξN , the anchoring strength required to induce a dual morphology will increase (i.e. ξS decreases), whereas at constant H, the dual morphology will be observed at lower anchoring strengths as SW increases. For an ideal hybrid nematic cell (HNC), with one homeotropic substrate and one planar substrate, it is possible for an inversion to occur between the eigenvalues, corresponding to the scalar order parameter S and the biaxiality η . This can only happen when the tensor order parameter is described in its general form (biaxial), and in regions where the scalar order parameter is low and the dominant direction is given by the biaxiality. This effect has been predicted and observed by several authors 27–31 . In our previous work, we accounted for such a phenomenon using a novel theoretically informed Monte Carlo relaxation method 32 , where the progression of the tensor order parameter was tracked at all times and its eigenvalues were kept in order. In addition, it is possible to obtain a critical thickness, around 30 to 50 nm, where the HNC exhibits a transition between the splay-bend alignment and uniform morphology, with the eigenvalues exchanged. In this work, we use a minimum thickness of 100 nm to avoid such a transition. For the systems considered here, we show that a characteristic length scale or ratio of length scales exists where a “sharp" transition in morphology is observed. We identify the


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conditions (SW , H and W ) and the physical mechanism (the roles of anchoring energy and the elastic deformations) where this transition between a uniform undistorted perpendicular alignment to a mixed uniform/splay-bend alignment occurs. The sharp transition that we report is of interest because it provides a route for the fundamental understanding of the LC morphological behavior due to independent anchoring patterns; knowledge that could be used to create low-power switching devices and to design the ground state of LC systems.

2

System

We consider a nematic LC confined between parallel plates (slit geometry) with a separation H. Homeotropic alignment is imposed at the top wall with an anchoring strength of WH . The “background” of the bottom wall is homeotropic, but it is decorated with degenerate planar stripes. These have width SW and anchoring strength WP . Depending on the values of H, SW and WP , an orientational transition is predicted from a uniform (U) alignment (perpendicular to the walls) to a nematic field that exhibits two stable alignments. This mixed regime consists of regions of uniform orientation and regions having a splay–bend (SB) “dome". Figure 1 provides a schematic representation of the system, and the two possible LC configurations that we anticipate. In what follows we examine the precise nature of the transition.

3

Analytical theory

We start by developing an analytical model of the LC free energy functional by recognizing that the mixed anchoring conditions may or may not impose an elastic distortion on the LC molecules. To do so, we build on the previous work by Berreman and Sambles 1,33 . Later in this manuscript we consider the case of elastic anisotropy. Here, however, we begin with a simple approximation, resulting in an expression for the Frank free energy density of the form: fE = K1/2 κθx2 + θy2 where κ = 1 − (1 − K2/K1 ) sin2 φ , K1 = K3 6= K2 are the splay, 5 ACS Paragon Plus Environment

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H ~ SW ~ ξN

H

z

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Homeotropic anchoring (WH)

x

Sw

SA Wp < Wp,c

Planar anchoring stripe (WP)

MA Wp > Wp,c

θ

SB dome

90 80

Twist line

70 60 50

θ

40 30 20 10

Uniform (U)

Uniform (U)

Splay-Bend (SB) Single Alignment (SA)

0

Mixed Alignment (MA)

Figure 1: Schematic representation of the LC slit system with height H. Homeotropic LC alignment is imposed at the top surface, while a homeotropic dominated anchored surface with a planar linear stripe of width Sw is placed at the bottom. The anchoring strength (WH or P ), the slit height (H) and the planar strip width (SW ) determine a transition between a single uniform alignment (SA) to a mixed, uniform/splay–bend, alignment (MA). Simulated contours of the LC alignment along the z–axis are included to illustrate the alignments and the splay-bend dome.


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bend and twist elastic constants and θ (x, z) is the LC alignment orientation field within the cell 1,33 ; boundary conditions are specified by the molecular alignment at the top and the patterned-bottom surfaces. Note that for a perfect hybrid cell, where a homeotropic anchoring is imposed at the top and degenerate planar anchoring is imposed at the bottom, the elastic distortions of the director field are dominated by splay and bend deformations. In this situation, even if the twist modulus is less than the splay and bend moduli, a twist deformation does not arise due to the nature of the boundary conditions 27 . In our system, on the other hand, we rely on a combination of planar and homeotropic anchoring alignments at the patterned surface. Therefore, there is an interface between the splaybend morphology and the uniform alignment. At this interface 34 , the twist deformation dominates when K2 is equal to or less than K1 , i.e. the LC molecules must accommodate the different orientations at the patterned surface; they need to transition from planar to perpendicular and, because the K2 is lower, the easiest thermodynamic mechanism is to undergo a twist. As explained in the Appendix, the total free energy, which accounts for finite anchoring conditions, is minimized with respect to the surface alignment, θ0 , resulting in K1 θ0 csch2 (H κψ ) (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ )) WP = 4Sw J1 (2θ0 ) 2 K1 θ0 csch (H κψ ) (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ )) . ≈ 2Sw sin [2(θS − θ0 )]

(1)

Here Jn (x) (n = 0, 1) are Bessel functions of the first kind. Note that if K = K1 = K2 then limSw →∞ WP = 2K θ0 / (H sin[2(θS − θ0 )]), which corresponds to the reported anchoring energy for a hybrid cell 35 . Figure 2 shows the θ0 −interval where J1 (θ0 ) and sin[2(π /2 − θ0 )] are similar. The anchoring dependence as a function of θ0 , for H = 100 nm and Sw = 200 nm, is also shown. This figure serves to illustrate how the anchoring diverges as the surface alignment approaches the easy axis. In the uniform phase, θ0 = 0 for every value of x; therefore, a critical anchoring strength,


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0.6 0.4 J1(2θ0)

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0.5

1.0

1.5

2.0

2.5

3.0

Theory Simulations

0.002 0.001 0.000

0.0 0.2 0.4 0.6 0.8 θ0

1.0 1.2 1.4

Figure 2: (Top) J1 (θ0 ) and sin[2(π/2 − θ0 )] as a function of the surface alignment θ0 . (Bottom) Anchoring dependence as a function of θ0 , it provides the anchoring energy for the uniform case and shows how the anchoring increases and diverges as θ0 approaches to the easy axis orientation (π/2 for planar anchoring).


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WP,C , can be estimated directly from Eqn. (9): WP,C = lim WP θ0 →0

K1 π csch2 (H κψ ) (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ )) = 4Sw

(2)

indicating the unique dependence of WP,C on a material parameter K and the geometry of the construct, H and SW . This analytical description provides a first order estimate of the free energy functional, and facilitates prediction of the critical anchoring strength that makes the LC start to “feel" the planar stripe. However, we emphasize that important effects are ignored by the simplifications within the theory. For instance, the LC alignment in the planar stripe, Eqn. (5), imposes a continuous change from a perpendicular orientation at x = ±SW /2 to a planar orientation at x = 0. This simplification results in three major consequences: an overestimation of the elastic distortions along the planar stripe, a constant surface penalization and a “naturally" imposed smooth transition between morphologies. In addition, the theory ignores the defect line, generated by the LC twist, along the homeotropic/planar interface at the bottom surface (see Fig. 1).

4

Numerical relaxation of the tensorial representation of the Landau-de Gennes free energy functional

For a more complete description of the system, we turn to numerical simulations of a tensorial free energy functional. In the Landau-de Gennes framework, it is expressed in terms of an alignment tensor field, Q = S [nn − δ /3] + η n′ n′ − (n × n′ )(n × n′ ) .


(3)

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Its eigenvalues determine the scalar order parameters, S and η (the biaxiality), that represent the level of order along the principal directions n and n′ (the eigenvectors). The thermodynamic state is defined by the following energy functional F(Q):

F(Q) =

Z

d 3 x [ fL (Q) + fE (Q)] +

I

d 2 x fS (Q),

(4)

where the short-range free energy density fL , the Landau density, captures (or defines) the isotropic-nematic (IN) transition, fE contains the long-range elastic distortions, and LC-surface interactions are quantified by a surface potential fS . In Doi’s notation 36 , the Landau free energy density fL includes two parameters that define the energy scale and determine the IN transition, namely A and U, respectively ( fL = fL (A,U, Q)). The elastic contributions are given in terms of the gradients of the alignment tensor. These gradients include elastic constants that may be related to the first order Frank-Oseen elastic constants ( fE = fE (Ls, Q, ∇Q) = fE (Ks, Q, ∇Q), where Ls are constants that may be expressed in terms of the elastic moduli, the Ks) 37–39 . Finally, surface potentials are introduced to describe the interaction of the liquid crystal with a surface. As mentioned earlier, for homeotropic anchoring we adopt a second order Rapini-Papoular 40 expression. Degenerate planar anchoring is included through the 4th order Fournier-Galatola 41 potential. Both expressions include an anchoring strength, WH or P , that determines how closely the surface alignment is followed by the LC molecules ( fS = fS (W, Q, ν ), where ν is the surface normal vector). The 4th order Fournier-Galatola potential was originally defined with two anchoring constants. In our representation, we considered that both constants are equal, W1 = W2 = WP . This approximation has been used previously 32,42–44 . The simulations are performed for a system representative of 4’-pentyl-4-cyanobiphenyl (5CB) liquid crystal, for which A = 1 × 105 J/m3 , K1 = 6 pN and K2 = 3 pN, resulting in a p nematic coherence length (NCL) ξN = L/A ≈ 7.15 nm. Experimentally, a wide range of 10 ACS Paragon Plus Environment

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NCL values have been reported for 5CB, depending of the type of experimental technique (X-ray, Neutron ray, NMR); reported values are between 10 to 50 nm 14,15,45,46 . Our selection is based on pure scaling principles in order to find and impose a characteristic length scale. Strong nematic conditions are imposed with U = 5 corresponding to Sbulk = 0.76. The homeotropic anchoring is assumed to be strong with WH = 1 × 10−2 J/m2 . The planar anchoring strength WP is varied from 1 × 10−4 J/m2 to 1 × 10−2 J/m2 until the mixed alignment is observed at constant H and SW . A Finite Element (FE) quadrature is used to discretize, integrate and calculate the free energy during theoretically informed Monte Carlo (MC) relaxations. Ten-noded isoparametric quadratic tetrahedral elements are used to approximate all variables and fields. The numerical integration is carried out during the MC following an O(1) algorithm; the quadratic elements allow us to achieve a high resolution level of approximation including sub-10 nm features generated from our nano-fabrication platform. Cubit (ver 14.1) 47 , used under the Argonne National Laboratory’s license, is used to generate the tetrahedral meshes. Furthermore, LibMesh 48 libraries are used to optimize and parametrize the original meshes. Additional details on the numerical method adopted in this work can be found in our previous publications 32,49 . The simulated cells measured 1.5 µ m × 1 µ m and had a thickness of 100 nm. Each system was discretized using ∼ 1.5 × 106 elements. The sub-10 nm features were resolved by ensuring that the average element size around them was 1 nm. During the Monte Carlo relaxation, at each configuration of the alignment tensor field Q(x) a probability is assigned proportional to the Boltzmann factor, i.e. exp[−β F(Q(x))] (where β −1 = kB T , kB is Boltzmann constant and T plays the role of a temperature). The Monte Carlo relaxation was carried out following an annealing process and using Metropolis sampling. This minimization method avoids entrapment in local minima along the underlying free energy landscape 32,42,49 . A typical simulation required between 9 × 106 and 1 × 107 accepted Monte Carlo steps per element. During this relaxation an annealing is performed where


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the inverse “fluctuating" temperature was varied from 10 to 10 × 1012 .

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8

Theory Simulation

6

SW = 420 nm

4 2 0 -5 10

SW = 295 nm SW = 215 nm WP,Exp

10-3 10-4 2 WP (J/m )

10-2

Figure 3: Total free energy per unit volume as a function of the anchoring strength WP for different stripe widths and H = 100 nm. Lines are for theoretical predictions, while symbols are for the simulations. Dotted lines depict the surface-dominated free energy and continuos lines corresponds to the elastic-dominated regime.

Figure 3 shows the total free energy per unit volume as a function of the planar anchoring strength for a cell thickness H = 100 nm and three different stripe widths. Analytical predictions are depicted by lines (continuous and dotted) and simulations are represented by the symbols. At low anchoring strengths, the LC prefers to ignore the presence of the planar stripe and follows a single uniform orientation. Therefore, surface penalizations dominate the free energy (dotted lines). As the anchoring increases, LC molecules prefer to generate elastic distortions, thereby forming the splay–bend deformation field and give rise to the mixed morphology. A critical anchoring strength WP,C is obtained at the point where the free energy changes from surface- to elastic-dominated. Both, our approximate theory and our detailed simulations predict the same qualitative behavior; given the sim-


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plifications in the analytical theory, however, the magnitude of the critical anchoring is underestimated. Note that as SW decreases, the required anchoring strength to generate the transition increases. This observation is highlighted in Fig. 4(Top), where WP,C is plotted as a function of SW for different slit heights. When the analytical predictions, from Eqn. (2), are multiplied by a factor Π = 18H −0.3 they overlap with the simulated predictions. This means that the underestimation by the first order theory is a unique function of H. Figure 4(Top) can be thought as an “alignment diagram": the lines (symbols) separate single uniform alignment (SA) region to the mixed uniform/splay–bend alignment (MA) region. The main conclusion from these results is that WP,C decreases as SW or H increases, and there is a unique relation between WP,C , SW and H that can be used for device design. Figure 4(Bottom) shows a variant of the results, where the elastic free energy is plotted as a function of H for a constant planar anchoring strength WP = 4 × 10−4 J/m2 (value that corresponds to the anchoring strength of our experimental validation below). Presenting the results in this way helps identify the critical planar stripe width where the single to multiple alignment (SA–MA) transition occurs. Single uniform alignments are characterized by ∇n = 0, whereas the appearance of the splay–bend orientation results in ∇n 6= 0. For instance, at H = 100 nm, a SW ∈ [300, 350] nm is felt by the LC creating the mixed alignment; on the other hand, for H = 150 nm, the critical width is SW ∈ [150, 200] nm. Conversely for SW > 300 nm, the mixed morphology is observed at every simulated slit height. From the thermodynamic point of view, the “sharp" morphological transition is predicted by simulated free energy functional in Fig. 3, where there is a clear discontinuous gradient in the functional at the point where the anchoring and elastic contributions balance each other. However, this sharp transition is identified directly in the elastic free energy in Fig. 4(Bottom). In addition, the analytical theory, complemented with the Π factor, may be transform to find the planar width where the dual morphology is observed as a function of the cell thickness. This expression is plotted in green in the figure for an anchoring strength WP = 4 × 10−4 J/m2 , delimiting the regions with single and mixed


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Theory x Π Simulation

5x10-4 4x10-4

MA

WP,C(J/m2)

H = 100 nm

Experiment

H = 200 nm

SA 10-4 150 4000

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200

H = 500 nm

250 300 SW (nm)

350

SW = 365 nm SW = 315 nm SW = 250 nm

Δn=0 (MA)

3000 2000

SW = 200 nm

1000 Δn=0 (SA)

0 100

200

Theory x Π (WP = 4x10-4 J/m2)

300 400 H (nm)

500

Figure 4: Analytical and simulated results for: (Top) the critical anchoring strength WP,C as a function of the planar stripe width SW and the cell thickness H. The lines delimit the UN–SB phase transition. The quantity Π = 18H −0.3 corresponds to factor by which the theory underestimates the results of simulations; and (Bottom) the elastic free energy as a function of H for different planar stripe widths with WP = 4 × 10−4 J/m2 . Uniform phases are characterized by FE = 0, whereas an splay–bend phase has FE 6= 0. With these anchoring conditions, planar stripes bellow 250 nm are able to induce a sudden uniform to splay-bend transition at a specific critical cell thicknesses.


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alignments.

5

Experimental Validation.

To validate our theoretical predictions, we used a slit cell with thickness H filled with 5CB. The confinement walls were made from two different substrates. For the top surface, an octadecyltrichlorosilane (OTS) modified glass was used to impart strong homeotropic anchoring. For the bottom surface, a silicon, coated with a poly(6-(4-methoxy-azobenzene4-oxy) hexyl methacrylate) (PMMAZO) polymer brush, was treated to generate controlled planar stripes on the homeotropic PMMAZO/silicon surface. E-beam patterning and O2 plasma etching were used to generate the chemical patterns. The removal of the polymer brush results in a 4 nm topological step between the homeotropic and the planar regions. Details of the experimental preparation method are provided in Ref. [ 34 ]. To estimate the anchoring strengths, we first determine the tilt angle from measurements of the retardance and the LC cell thickness. Once the tilt angle is known, the anchoring strength of the PMMAZO brush on the silicon substrate and the oxygen plasma treated silicon are calculated using W = 2K(θ0 − θOTS )/(H sin[2(θe − θ0 )])

35 ;

they are

WP = 4.2 ± 2.0 × 10−4 J/m2 for the planar stripes, WH = 3.2 ± 1.0 × 10−5 J/m2 for the homeotropic PMMAZO brush areas, and WH = 1 ± 0.9 × 10−2 J/m2 for the homeotropic OTS substrate 50–55 . Multiple stripes were patterned having a total length of 80 µ m and a separation of 5 µ m between them, to ensure that end effects and inter-planar interactions are negligible. The width of the patterns (planar regions) was controlled systematically from 150 nm to 400 nm, while the slit height was kept at H = [100, 160, 280, 370, 540] nm. The characterization of the nematic field orientation, the single uniform alignment and the mixed uniform/splay–bend morphology, were performed by placing the LC cell between crossed polarizers and by measuring the brightness intensity (BI). A BX-60 Olympus polarized light microscope was used with a white light source where the lamp was set at 12 V/100 w; the exposure time was 90 ms. ImageJ 56 was used to process the intensity


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images, where an image stack was generated for each H and a single brightness intensity adjustment was done for SW = 400 nm to eliminate the optical noise between experiments and to limit the maximum BI to a single value for every H. Figure 5 shows the brightness intensity as a function of the planar stripe width SW and the cell thickness H. The brightness is normalized with the stripe width to discard volumetric changes in the measurement. The cross polarizers are oriented perpendicular to each other at an angle of 45o with respect to the planar stripes (green line). Reflective light microscopy images are included in the figure for four widths and four cells. Recalling that WP = 4.2 ± 2.0 × 10−4 J/m2 , our calculations predict a single-to-mixed morphology transition between SW = 300 and 350 nm for H = 100 nm and between SW = 150 and 200 nm for H = 160 nm (see Figs. 3 and 4). This behavior is verified by our experiments, where a sudden increase of the brightness at SW > 300 nm is observed for H = 100 nm, and at SW > 150 nm it is observed for H = 160 nm. For the thicker cells, the brightness intensity remains almost constant. In the figure, we include simulated brightness calculations for H = 100 and H = 540 nm; the theoretical reflected light images are in good agreement with our experimental observations. To demonstrate the usefulness of the concepts discussed here, a master cell was prepared with thickness H = 100 nm, containing arrangements of linear stripes with SW ranging from 150 to 500 nm, and a total area of 120 µ m × 195 µ m. An electric field perpendicular to the walls, i.e. z–direction, was applied using a voltage difference of 4.5 V, and the time required for the brightness to vanish was recorded for each stripe width. Figure 6 shows the response time as a function of SW and reflective light microscopy images at time zero, 30 and 60 s after the field is applied. For stripe widths that follow a single uniform alignment (SW < 300 nm), the electric field must orient only the residual molecules near the planar surface, and the time required is short (∼ 25 s). These residual molecules are responsible for the low brightness shown in Fig 5 for SW < 300 nm. On the other hand, for stripes above 300 nm the electric field must re-orient the molecules in the the splay-bend


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45o

BI/SW (Intensity/nm)

0.25

H = 540 nm

0.2

H = 370 nm

0.15

H = 280 nm 0.1

H = 160 nm

0.05 0 150 H

SW

540 nm 370 nm 160 nm 100 nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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200

150 nm

H = 100 nm

250

300

SW (nm)

200 nm

350 nm

350

400

400 nm

45o

10 μm

Figure 5: Brightness intensity (BI), divided by the planar stripe width to discard volumetric effects, and reflective light microscopy images under the crossed polarizers, for different cell thickness, H, as a function of the planar stripe width SW . The red insets for H = 100 and 540 nm are the simulated brightness intensity.


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dome, resulting in longer times. As SW is increased above the critical stripe width for the single-to-mixed morphology transition the re-alignment time increases monotonically. Recall that the time is measured according to the brightness that excludes the effects of the different volumes on the optical signal. An increasing time implies that it is “easier" to reorient the LC molecules if the stripe width is closer to the “critical” transition magnitude. It is then possible to engineer an LC cell using a specific transition time as the design parameter, where the chemical patterns close to the morphological transition would ensure that a lower voltage be used to induce the transition. Giving the nanoscale control afforded in our platform, an entire device could be fabricated with different transition zones (time or voltages) or, more importantly, with low-power consumption characteristics. Note that the experiments in Fig. 6 were carried out at room temperature (T = 25o C), where the 5CB is in a nematic phase. At this temperature, the critical voltage of the Freédericksz’ transition in the planar anchoring state is estimated to be 1.0 V 57 . This value is defined as the voltage to induce an infinitesimal deformation of the director field. It is expected that the voltage to enter a completely dark state is higher than the critical voltage. As such, 4.5 V is a reasonable value to achieve a completely dark state. The slow dynamics of the director field, however, is explained by the flow of the liquid crystal. The 5CB materials we used is 98% without further purification; we assume that some charged particles remain in the materials. An applied electric field will also cause the flow of those particles, thereby applying a torque to the LC molecules that surround the particles, and induce the LC molecules to be rearranged. In that case, the LC molecules would slowly realign under the influence of the particles in the system.

6

Conclusions

A combined theoretical and experimental approach was used to examine the response of LCs to nano-patterned substrates using polymer brushes. It was found that if the dimensions of the system are comparable to the characteristic dimensions of the LC, an abrupt


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90

H = 100 nm

80 Time to BI/SW = 0 (s)

70

t=0

60 50 40 30 20 200

250 nm

300

350 SW (nm)

300 nm

350 nm

450

500

400 nm

t = 30 s

t =0

SW

250

t=0 400

t = 60 s

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Figure 6: Time required for BI/SW to be zero in a cell with H = 100 nm and an applied electric field perpendicular to the walls (z–direction) with a voltage difference of 4.5 V. Reflective light microscopy images under the crossed polarizers are included for several stripe widths at t = 0, 30 and 60 s after the field in turned on. The patterns are 120 µm × 195 µm.


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orientational transition may be induced and controlled by the surface chemistry and geometric design. This transition is observed from an undistorted uniform alignment to a mixed morphology consisting of regions with uniform alignment interdispersed with regions of splay–bend orientation. The SB domains arise from the local combination of planar and perpendicular anchoring at the walls, giving rise to interesting elastic distortions that may be used as a scaffold for nano-particle directed assembly. By creating patterned surfaces with dimensions in the immediate vicinity of the characteristic length scale for the observed transition, it was shown that the resulting hybrid devices exhibit faster response times to applied voltage. Taken together, the results presented in this work provide a simple proof of concept demonstration suggesting that patterning techniques that have thus far been applied in the context of commercial block copolymer assembly may be used for development of fast and efficient LC optical devices.

7

Appendix

The minimization of the elastic energy results in a harmonic orientation field, κθxx + θzz = 0, with homeotropic boundary conditions at the top surface that impose a homogeneous Dirichlet condition at z = H, θ (x, H) = 0, while the bottom surface is represented by    θ0 cos(π x/Sw ) if kxk ≤ Sw 2 θ (x, 0) =   0 otherwise

(5)

where θ0 is the angle of the director field at x = 0 and z = 0 and SW is the planar stripe width. We assume that the orientation of the nematic field is constant along the y–axis. The harmonic orientation field, with the corresponding boundary conditions, is obtained as an eigenfunction expansion, i.e. zκψ Hκψ sinh(zκψ ) θ (x, z) = θ0 cos(xψ ) e +e , sinh(H κψ )


(6)

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where ψ = π /Sw . Recall that solutions for θ (x, z) in Eqn. (6) minimize the elastic distortions that are imposed by the surface anchoring. However, the configuration that minimizes the total free energy must account for the finite anchoring conditions. Using a second order Rapini-Paupolar potential for the surface interactions, the total free energy functional (per unit length along the y–axis) may be written as ˆ θ ) = K1 F( 2

Z HZ

− S2w

0

WP 2

Sw 2

Z

Sw 2

− S2w

κθx2 + θy2 dxdz+

(7)

2

sin (θ (x, 0) − θS ) dx,

where WP is the anchoring strength at a planar stripe. Note that the LC at a homeotropic surface is assumed to follow a perpendicular orientation (i.e. there is no surface contribution and the anchoring strength is assumed to be infinite). The surface orientation θS is the orientation of the easy axis which, for planar anchoring, corresponds to π /2. To obtain an expression for WP , the free energy functional in Eqn. (7), assuming the orientation field in Eqn. (6), namely

F(θ ) =

K1 ψθ02 csch2 (H κψ )× 16 (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ ))

(8)

1 + WP Sw (1 + J0 (2θ0 )), 4 is minimized with respect to the surface alignment θ0 , resulting in WP =

K1 θ0 csch2 (H κψ ) (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ )) 4Sw J1 (2θ0 ) K1 θ0 csch2 (H κψ ) (2H πκ (κ − 1) + Sw (κ + 1) sinh(2H κψ )) ≈ , 2Sw sin [2(θS − θ0 )]

(9)

where Jn (x) (n = 0, 1) are Bessel functions of the first kind. In order to ascertain how a specific value of the homeotropic anchoring strength may


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affect the critical planar anchoring (WP,C ), we have carried out a sensitivity analysis. The results are shown in Fig. 7. We simulated several scenarios, where the homeotropic anchoring strength was decreased by one and two orders of magnitude, respectively, at the top surface and at the bottom patterned substrate. In the figure, we plot the total free energy per unit volume as a function of the planar anchoring strength WP for a cell of thickness H = 100 nm. We considered three different planar stripe widths SW = 215, 295 and 420 nm. For these simulations, we distinguish the homeotropic regions at the top Top

and bottom substrates, to better mimic the experimental system. We define the WH

and

WHBottom . The values for the top anchoring strength are in the strong regime, i.e 10−2 and 10−3 J/m2 , and in the moderate regime 10−4 J/m2 . In Fig. 7, the results shown on the left column are for different values of WHBottom . For the results on the right column, WHBottom was fixed at 10−5 J/m2 to reproduce better the experimental conditions. In the figure, we are also including the relative difference in the location of the morphological transition beTop

tween WH

= 10−2 J/m2 and the other two values 10−3 J/m2 and 10−4 J/m2 . We find that

for any combination of homeotropic anchorings, the location of the morphological tranTop

sition - once WH

is in the strong anchoring regime - is within a 4% difference between

10−2 J/m2 and 10−3 J/m2 . Decreasing by one order of magnitude the anchoring for the homeotropic region leads to equivalent predictions. These results reinforce our hypothesis that, when strong anchoring conditions are used at the top, the morphological transition depends almost exclusively on H, WP and SW . For completeness we have also shown that, once WH is in the moderate anchoring regime, i.e. 10−4 J/m2 , the value of the critical planar anchoring strength, WP,C , decreases by about 20% to 30% when compared with the strong anchoring regime.


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Bottom

F/V (kBT/nm3)

10 8 6

Relative difference

10 0 -4 10

SW = 215 nm

20 (%) Relative difference

10 0 -4 10

10-3 10-2 WP (J/m2)

10-3 10-2 WP (J/m2)

Top

Top

WH =10-4 (J/m2)

WH =10-3 (J/m2)

Bottom

Bottom

WH =10-3 (J/m2)

10 F/V (kBT/nm3)

WH =10-5 (J/m2)

SW = 215 nm

20 (%)

4

20

0 -4 10

WH =10-5 (J/m2)

SW = 295 nm

(%)

8 10 6

Bottom

WH =10-2 (J/m2)

2

Relative difference

SW = 295 nm Relative difference

10 0 -4 10

10-3 10-2 WP (J/m2)

4

(%) 20

10-3 10-2 WP (J/m2)

Top

WH =10-2 (J/m2)

2 Bottom

10

F/V (kBT/nm3)

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Bottom

WH =10-4 (J/m2)

WH =10-5 (J/m2)

SW = 420 nm

SW = 420 nm

8

Top

WH

6 20

4 2 10-5

SW

(%)

10 0 10-4

Relative difference

10-3

10-4 10-3 WP (J/m2)

Bottom

WH

H WP

10 0 -4 10

10-2

10-2 10-5

(%) 20

Relative difference

10-3

10-4 10-3 WP (J/m2)

10-2

10-2

Figure 7: Total free energy per unit volume as a function of the planar anchoring strength WP for H = 100 nm and three different stripe widths (SW ): 215 nm, 295 nm and 420 nm. For the left column, the bottom homeotropic anchoring strength was varied from 10−2 J/m2 to 10−4 J/m2 . For the right column, WHBottom was kept at 10−5 J/m2 . Insets show Top the relative difference in the location of the morphological transition between WH = Top Top 10−2 J/m2 and the two other values: WH = 10−3 J/m2 and WH = 10−4 J/m2 . In the Top figure, black lines and symbols are for WH = 10−2 J/m2 , red lines and symbols are for Top Top WH = 10−3 J/m2 and blue lines and symbols are for WH = 10−4 J/m2 .


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Acknowledgement The authors acknowledge support from the Department of Energy, Basic Energy Sciences, Biomaterials Program under Grant No. DE-SC0004025. Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program of the Argonne Leadership Computing Facility at Argonne National Laboratory. Additional development work was performed using the Argonne Laboratory Resource Computing Center (LCRC) and the University of Chicago Midway cluster. JCA-P and JAM-G are thankful to CONACYT for the Postdoctoral Fellowships Nos. 186166 - 203840 and 250263, respectively. JPH-O is grateful for funding provided by the Universidad Nacional de Colombia Ph.D. grant and COLCIENCIAS under the Contract No. 110-165-843-748, “Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, Tecnología y la Innovación Francisco José de Caldas."

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Ignoring the Pattern

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H11C5

CN

45o

Planar anchoring ACS Paragon Plus Environment Homeotropic anchoring

Feeling the Pattern

Sharp Morphological Transitions from Nanoscale ... - ACS Publications

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