Chevron structures in smectic A liquid crystals

face anchoring at the plates does not allow the adjacent lay-ers to rotate freely. As a result, the layers undulate with the tilt angle changing sign ...

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Chevron structures in smectic A liquid crystals Tiziana Giorgi New Mexico State University Thursday 26th November, 2015

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Ideal picture for common liquid crystalline (thermotropic) mesophases in elongated rod-like molecules A preferred local average direction can be defined, represented by a unit vector, n - the director

 Nematic phases: no positional order, but some orientational order  Smectic phases: some orientational order, and positional order  Common smectic phases: Smectic A with layers perpendicular to the director; Smectic C with director tilted with respect to the layers

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RAPID COMM

PHYSICAL REVIEW E, VOLUME 63, 030501共R兲

Undulations in a confined lamellar system with surface anchoring T. Ishikawa and O. D. Lavrentovich Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242 共Received 16 February 2000; published 27 February 2001兲 We visualize undulations in layered systems using a cholesteric stripe phase with a macroscopic supramicron periodicity. The wave vector of stripe pattern is in the cell’s plane. The undulation is induced by an in-plane magnetic field normal to the stripes. The observed displacement of layers is much larger than the value predicted by the Helfrich-Hurault classic theory. We propose a model of undulations that explains the data by RAPID COMMUNICATIONS finite surface anchoring of layers.

HIKAWA ANDDOI: O. D.10.1103/PhysRevE.63.030501 LAVRENTOVICH

I. INTRODUCTION

PHYSICAL PACS REVIEW E 63 030501共R兲 number共s兲: 61.30.⫺v

II. EXPERIMENT

The model system with an undulating stripe pat A variety of condensed phases possess reduced onecreated in two steps: 共i兲 obtaining a uniform choleste dimensional 共1D兲 共smectic兲 or 2D 共columnar phases兲 transgerprint texture 关16兴; 共ii兲 generation of undulations by ational order that allows long-range curvature deformations netic field in the plane of the cell. splay in smectic, bend in columnar phases兲 关1兴. Curvature 共i兲 The cell is assembled from a pair of glass plates deformations are capable of relaxing dilation or field-induced with transparent 共ITO兲 electrodes and an alignment m tress. In many systems 共smectic Awikipedia 关2,3兴, cholesteric 关4兴 FIG. and 4. Undulation pattern near the mylar wallRubber兲 (H⫽1.05Hthat JALS 214 共Japan Synthetic c ). sets home columnar 关5,6兴 liquid crystals, diblock copolymers 关7,8兴, peboundary conditions. Two mylar strips are placed b iodic patterns in ferrofluids 关9兴,nematic and ferrimagnets 关10,11兴, i.e., the function u 0 (H/H on oneother, material cholesteric: chiral LC the glass plates parallelonly to each separated by c ) depends 冑K/B by the the cell. ratio The myla the elastic lengthmm ␭⫽in etc.兲, the dilation-curvature coupling shows up as the parameter, undutance a⫽1.7 the defined plane of p: cholesteric pitch of the curvature K to the l⫽(15.7⫺16) compression modulus ation instability, often also called buckling or Helfrich␮ m B(⬇ofP) betwee fix constant the distance

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PHYSICAL REVIEW E 74, 011712 共2006兲

Undulations of lamellar liquid crystals in cells with finite surface anchoring near and well above the threshold B. I. Senyuk, I. I. Smalyukh, and O. D. Lavrentovich* Liquid Crystal Institute and Chemical Physics Interdisciplinary Program, Kent State University, Kent, Ohio 44242, USA 共Received 19 April 2006; published 31 July 2006兲

We study the undulations instability, also known as the Helfrich-Hurault or layers buckling effect, in a cholesteric liquid crystal confined between two parallel plates and caused by an electric field applied along the normal layers. The LIQUID cholesteric pitch is much but sufficiently large for optical NDULATIONS OF to LAMELLAR CRYSTALS IN¼smaller than the cell thickness PHYSICAL REVIEW E 74, 011712 共2006兲 study. The three-dimensional patterns of the undulating layers in the bulk and at the surfaces of the cells are determined by fluorescence confocal polarizing microscopy. We demonstrate that the finite surface anchoring at the bounding plates plays a crucial role in the system behavior both near and well above the undulations threshold. The displacement of the layers immediately above the undulation threshold is much larger than the value expected from the theories that assume an infinitely strong surface anchoring. We describe the experimentally observed features by taking into account the finite surface anchoring at the bounding plates and using Lubensky-de Gennes coarse-grained elastic theory of cholesteric liquid crystals. Fitting the data allows us to determine the polar anchoring coefficient W p and shows that W p varies strongly with the type of substrates. As the applied field increases well above the threshold value Ec, the layers profile changes from sinusoidal to the sawtooth one. The periodicity of distortions increases through propagation of edge dislocations in the square lattice of the undulations pattern. At E ⬇ 1.9Ec a phenomenon is observed: the two-dimensional square lattice of undulations transforms into the one-dimensional periodic stripes. The stripes are formed by two sublattices of defect walls of parabolic shape. The main reason for the structure is again the finite surface anchoring, as the superposition of parabolic walls allows the layers to combine a significant tilt in the bulk of the cell with practically unperturbed orientation of layers near the bounding plates. DOI:E 10.1103/PhysRevE.74.011712 = 1.04 Ec

PACS number共s兲: 61.30.Eb, E = 1.4 Ec 61.30.Hn, 61.30.Jf

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Sawtooth structure in SmA by magnetic field Joint work with Carlos J. García-Cervera and Sookyung Joo Consider a LC sample in the smectic A phase confined between two flat plates with strong anchoring condition at the boundary plates. If the molecules are uniformly aligned so that the smectic layers are parallel to the bounding plates, a magnetic field applied in the direction parallel to the layer will tend to reorient the molecules and the layers, while the surface anchoring condition at the plates will oppose this reorientation

Picture: Encyclopædia Britannica

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Experiments show that in this setup, an instability occurs above a threshold magnetic field, where layer undulation appears (Helfrich-Hurault effect). This transition has been analyzed in great details by García-Cervera and Joo in previous works As the applied field increases well above this first critical value, the sinusoidal shape of the smectic layer changes into a sawtooth (zigzag) (also periodic) pattern with a longer period Here, we are interested in the higher field regimes, in particular in the description and derivation of the zigzag profile

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P.G. de Gennes: smectic layering description via order parameter To describe the smectic layering, de Gennes proposed a free energy, where in addition to the director n, we have an order parameter ψ, corresponding to the distribution of the centers of mass of the molecules: ψ(x) = ρ(x)e i q ω(x) ρ is the mass density of the smectic layers The smectic layers are the level sets of ω If ψ = 0 the molecules have no positional order, the centers of mass are distributed uniformly, and the liquid crystal is in the nematic phase If ψ is not identically equal to 0, the phase is smectic 1 is proportional to the layer thickness, λ q

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Modified Chen-Lubensky energy In our work, we use the modified Chen-Lubensky model (based on the de Gennes model for SmA, and introduced to investigate the nematic to SmA or SmC transition), and assume sample: (−L, L)2 × (−d, d) magnetic field applied in the x direction: h = e1 . ρ = constant, since we are interested in the development of the layers when these are well defined Under these assumptions, after non-dimensionalizing with respect to the thickness d of the sample, the energy we study can be written as follows: Z  dK D2 CL GC (ϕ, n) = D1 ε(∆ϕ − ∇ · n)2 + |∇ϕ − n|4 ε (−r ,r )2 ×(−1,1) 2ε  1 + |∇ϕ − n|2 + ε|∇n|2 − τ (n · h)2 d¯ x, ε in here ϕ represents the location of the layers, τ the strength of the magnetic field, r = L/d, and all the coefficients are positive.

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The dimensionless parameter ε is the ratio of the layer thickness λ, to the sample thickness d, and thus ε  1 (in the physics literature, one finds values such as d = 1 mm & λ = 20 Å) The parameter ε is used in the work of García-Cervera and Joo, where they investigate the first instability of GC , and find that the critical field, τc , at which undulations appear is of order τc = O(1) The sawtooth profile is instead observed experimentally at higher fields of order τ = O(ε−1 ) Therefore, we set σ = τ ε and treat σ as a constant

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One-dimensional case We start by considering the case n ∈ S1 , and assume ϕ = y − g (x), where g denotes the layer displacement from the flat position Setting n = (sin θ, 0, cos θ), with θ = θ(x), and I = (−r , r ), we have Z h D2 0 1 FεC (θ, g ) = D1 ε(g 00 + cos θ θ0 )2 + (g + sin θ)4 + (g 0 + sin θ)2 2ε ε I CL_S1

+

i 4D2 0 θ 1 (g + sin θ)2 sin4 + ε θ02 + W (θ) dx, ε 2 ε

where W (θ) = 8D2 sin8

θ θ θ + 4(1 + σ) sin4 − 4σ sin2 + a0 2 2 2

(a0 chosen so to ensure that W (θ) is nonnegative)

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Direct computations show that for θ ∈ [−π, π], W (θ) presents a double well potential (see picture below for D2 = 1)

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We denote the zero set of W by {±β} for β > 0, and define YC = {(θ, g ) ∈ W 1,2 (I ) × W 2,2 (I ) : θ(−r ) = θ(r ), g (−r ) = g (r ), g 0 (−r ) = g 0 (r )}, AC = {(θ, g ) ∈ BV (I ) × W 1,2 (I ) : g 0 ∈ BV (I ), 0

Z

g + sin θ = 0 a.e,

θ = 0} I

We let GεC (θ, g )

( FεC (θ, g ), := +∞

if (θ, g ) ∈ YC , else,

and (R

˜ )) − Φ(θ(−r ˜ |(Φ ◦ θ)0 | + |Φ(θ(r ))| if (θ, g ) ∈ AC , +∞ else Rs p ˜ where Φ(s) = 2 −β W (t) dt and θ(±r ) denotes the trace of θ on ±r Z Note that for (θ, g ) ∈ AC : |(Φ ◦ θ)0 | = Φ(β)(number of jumps) G0C (θ, g )

:=

I

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Using Γ-convergence, for  going to zero, we recover a zigzag profile. In particular, we have the following result: Theorem Let {(θj , gj )} ∈ YC be a sequence of minimizers to GεCj for εj → 0. Then there are sequences of numbers {aj } ⊂ {±1} and {cj } ⊂ (−r , r ) such that (θˆj , gˆj ) → (θ, g ) in L1 (I ) × L2 (I ) where θˆj (x) = aj θj (x + cj ), θ = βχJ − β(1 − χJ ),

gˆj (x) = aj gj (x + cj ),

with J = (−r /2, r /2). R s p Furthermore, G0C (θ, g ) = 2 Φ(β) ≡ 2 −β W (t) dt.

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Some numerics Numerical approximation of gradient flow equations associated with the Chen-Lubensky: Rectangular domain: (−r , r ) × (−1, 1) where r = 4 Dirichlet boundary condition on n = e2 and ϕ = y at y = ±1 and periodic boundary conditions at x = ±r (experiments show periodic patterns) The dimensionless parameters used are D1 = 0.1, D2 = 0.76, ε = 0.2 In the first row: we depict the configuration of each component of n in the middle of the domain, y = 0 – the numerics show how the undulatory pattern transforms to the zigzag pattern of the director In the second row: we show the layer description given by the contour map of ϕ – in the middle of the domain the layer profile changes from sinusoidal to sawtooth shape and its periodicity increases as the field strength increases

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Note that the first instability from the undeformed state (ϕ = y , n = e2 ) is observed as layer undulations at τ = π (first column)

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Recall: in our analysis we let ϕ(x, y ) = y − g (x) and look at the minimizers of the energy for g and n A numerical simulations that directly finds a minimizer of the energy in the simpler setting D1 = D2 = 0, by using a Truncated-Newton algorithm for energy minimization with a line search, and a Fourier spectral discretization in the x direction, gives analogous patterns: 1

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Figure : D1 = D2 = 0, A ≡ (1 + σ)−1 = 0.5 and (a) ε = 1 and (b) ε = 0.2.

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The numerics capture the essential feature of the zigzag pattern in the simplified setting D1 = D2 = 0 This is particularly interesting, since if we pick D1 = D2 = 0 in the original full Chen-Lubensky energy, we obtain   Z dK 1 G(ϕ, n) = |∇ϕ − n|2 + ε|∇n|2 − τ (n · h)2 d¯x, ε (−r ,r )2 ×(−1,1) ε which is what de Gennes energy for SmA looks like under our assumptions Suggesting that the de Gennes model might be enough to investigate the sawtooth formation phenomenon seen in experiments

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Two-dimensional case for de Gennes energy We study the layer structure in the cross section of the sample (z = 0), so that the problem is reduced to a two dimensional case i.e. take n = (n1 , n2 , n3 ) ∈ S2 , where n = n(x, y ), and as before h = e1 Setting ϕ = z − g (x, y ), the de Gennes energy gives   Z 1 1 1 ε|∇n|2 + W (n) + (gx + n1 )2 + (gy + n2 )2 dx dy , ε ε ε (−r ,r )2 where W (n) = σn22 +

1 (n3 − A)2 , A

with A = (1 + σ)−1 < 1. We denote the zeros of W : S2 → [0, ∞) by and write α = arccos(A)

n± = (±¯ n1 , 0, A),

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The mathematical analysis we adopt for the two dimensional free energy is motivated by the study of domain walls in ferromagnetism By reformulating the free energy, we have captured a double well potential having two minimum states for the director on the sphere We can then follow Anzellotti-Baldo-Visintin (1991), and use a Modica-Mortola-type inequality on the sphere equipped with a new metric associated with the double well potential Additionally, since experiments show periodic chevron patterns, we assume periodic boundary conditions, and adapt to S2 -valued vector fields, the variational approach on the flat torus presented in Choksi-Sternberg (2006), where they consider the Cahn-Hilliard energy in the periodic setting to study microphase separation of diblock copolymers While extending the techniques in Anzellotti et al. to the flat torus, we need to consider the presence of the smectic order parameter, and work with the explicit formula for a geodesic curve, in the new metric, connecting the minima of the well potential of the de Gennes energy

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We consider a two dimensional flat torus T2 = R2 /(2r Z)2 (roughly speaking: the square [−r , r )2 with periodic boundary conditions), and work with the energy:  Z  1 1 1 2 2 2 Fε (n, g ) = ε|∇n| + W (n) + (gx + n1 ) + (gy + n2 ) dx dy ε ε ε T2 where

W (n) = σn22 +

1 (n3 − A)2 , A

Γ-convergence setup: Y = W1,2 (T2 , S2 ) × W 1,2 (T2 ) A =

n (n, g ) ∈ BV(T2 , {n± }) × W 1,2 (T2 ) : gx ∈ BV (T2 , {±¯ n1 }), Z o g = g (x), n = n(x), gx + n1 = 0 a.e., n2 = 0 a.e., n1 = 0 T2

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and functionals, Gε and G0 , defined on X = L1 (T2 , S2 ) × L2 (T2 ): ( Fε (n, g ) if (n, g ) ∈ Y Gε (n, g ) := +∞ else ( 2c0 PT2 (An− ) if (n, g ) ∈ A G0 (n, g ) := +∞ else

PT2 (An− ) is the perimeter of the set

An− = {(x, y ) ∈ T2 : n(x) = n− },

while c0 = inf

nZ

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p

W (γ(t))|γ 0 (t)|dt : γ ∈ C 1 ([0, 1], S2 ), γ(0) = n− , γ(1) = n+

0

The value of c0 is given by

c0 =

√2 (sin α A

− α cos α)

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We have compactness, and G0 = Γ- limε→0 Gε . Then, the uniqueness of the pattern of the minimizers of G0 allows us to apply the method of proof used by Choksi-Sternberg (2006) to study a similar interface limit for a periodic system, and obtain the analogous of the one-d theorem: Theorem Let {(nε , gε )} ∈ Y be a sequence of minimizers of Gε . Then there exists a sequence {cε } ⊂ (−r , r ) such that (˜ nε , g˜ε ) → (n, g )

in L1 (T2 , S2 ) × L2 (T2 ),

where ˜ ε (x, y ) = nε (x + cε , y ), n n = n− χL− + n+ χL+ , L− = {x :

r 2

g˜ε (x, y ) = gε (x + cε , y ),

gx = −n¯1 χL− + n¯1 χL+

< |x| < r } and L+ = {x : |x| < 2r }.

Furthermore, G0 (n, g ) = 8 c0 r .

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Numerics García-Cervera-Joo (2012) studied the layer undulation phenomena in a two dimensional domain: (−r , r ) × (−1, 1) and n ∈ S1 They showed that the first instability occurs at τ = O(1), and when Dirichlet conditions are imposed on φ at z = ±1, they proved that the critical fields for undulational instability is π Here, the numerics are for a three dimensional domain with n ∈ S2 . One can show that in this case the same estimate for the critical field and description of the layer undulations can be obtained The first figure confirms that the layer undulation occurs at π: the numerical simulations show that undeformed state (n = e3 , φ = z) is an equilibrium state at τ = 3 and undulations appear at τ = 3.2 The sinusoidal oscillation transforms into chevron structures at a much stronger field as shown in the second figure. The numerical experiments also indicate that the period becomes larger as the field strength increases

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Figure : Undulations: The first and second rows depict the scalar components of directors, and the surface of the layer in the middle of the cell, respectively. The magnetic field strength τ = 3, 3.2, 7 for each column.

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Figure : Chevron structures: The first and second rows depict the scalar components of directors, and the surface of the layer in the middle of the cell, respectively. The magnetic field strength τ = 8, 13, 13.5 for each column.

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Numerics Numerical approximation of gradient flow equations associated with the de Gennes energy for a three dimensional domain and n ∈ S2 : a Fourier spectral discretization in the x and y directions a second order finite differences in the z direction temporal discretization: a combination of a projection method for the variable n, with a semi-implicit scheme for φ Initial condition: small perturbation from the undeformed state Boundary conditions: strong anchoring for n and Dirichlet for φ at the top and the bottom plates, that is for all t n(x, y , ±1, t) = e3 ,

and φ(x, y , ±1, t) = z

Periodic boundary conditions for both n and φ in the x and y directions For r = 4, ε = 0.2 and 128 grid points in the x, y and z directions, we obtain the following picture:

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Chen-Lubensky: two-dimensional case With n = n(x, y ) ∈ S2 , ϕ = z − g (x, y ) and h = e1 , denoting nk = (n1 , n2 ), the Chen-Lubensky energy becomes Z   D2 D1 ε(∆g + ∇ · n)2 + |∇g + nk |4 + 2(1 − n3 )2 |∇g + nk |2 2ε (−r ,r )2  1 + |∇g + nk |2 + ε|∇n|2 + W (n) dx dy , ε where W (n) =

D2 (1 − n3 )4 + (1 − n3 )2 + σ(n32 + n22 ) + b0 , 2

b0 chosen so to ensure that W is nonnegative

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W (n) is again a double well potential with two zeros n± = (±¯ n1 , 0, B), here r B =1−2

and n¯1 =



" 1 1+σ sinh arsinh 3D2 3

3σ 2(1 + σ)

r

3D2 1+σ

!# ,

1 − B2

And, one can se that B(σ, D2 ) → A = (1 + σ)−1 as D2 → 0+ A similar Γ-convergence analysis for this model results in a picture analogous to the one obtained for the de Gennes energy In particular, the same proofs give that the Γ-limit established for the de Gennes energy is also the Γ-limit of the Chen-Lubensky model, with A = (1 + σ)−1 replaced by B