Supporting Online Material for

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1 Supporting Online Material for Chiral Isotropic Liquids from Achiral Molecules L. E. Hough,* M. Spannuth, M. Nakata, D. A. Coleman, C. D. Jones, G. Dantlgraber, C. Tschierske, J. Watanabe, E. Körblova, D. M. Walba, J. E. Maclennan, M. A. Glaser, N. A. Clark* *To whom correspondence should be addressed. (L.E.H.); (N.A.C.) Published 24 July 2009, Science 325, 452 (2009) DOI: /science This PDF file includes: Materials and Methods SOM Text Figs. S1 to S9 References

2 1 METHODS CONTENTS 1a - Freeze Fracture Transmission Electron Microscopy (FFTEM) 1b - X-Ray Diffraction (XRD) 1c - Optical Rotation (OR) 2 NOTES 2a - Relation to Pasteur s Experiment 2b - Electro-Optics 2c - Electro-Optics of Layer Stackings Consistent with Curvature 2d - Focal Conic Defects as a Model for the Dark Conglomerate Layer Structure 2e Elastic Modeling of Dark Conglomerate Layer Deformation 3 - SUPPORTING FIGURES S1-S9 4 - SUPPORTING REFERENCES - 1 -

3 1 - METHODS 1a - Freeze Fracture Transmission Electron Microscopy (FFTEM) In previous studies of smectic liquid crystals (S1, S2) and bent-core molecular systems (S3), freeze fracture TEM (FFTEM) has enabled visualization of the smectic layer organization. Our FFTEM experiments were carried out by sandwiching the LC between 2mm by 2mm glass or copper planchettes and cooling at several rates to selected temperatures in the DC range. The phase was confirmed by optical microscopy in the case of glass planchettes. The samples were then rapidly quenched to T< 90 K by immersion in liquid nitrogen or propane, and fractured cold in a vacuum. TEM of Pt-C fracture face replicas revealed the topographic interface structure of the fracture, which occurs preferentially parallel to the layer surfaces. Layer surfaces can be identified unambiguously by occasional layer steps. Nylon coating of the glass was found to preferentially orient the layers of some materials with the layer normal parallel to the glass. With this orientation, if the fracture surface passes near the LC/glass interface, the layers intersect the fracture plane at right angles thereby revealing their organization in the surface. 1b - X-Ray Diffraction (XRD) XRD experiments on powder samples were carried out on a Huber four-circle goniometer on beamline X10A of the National Sychrotron Light Source at Brookhaven National Laboratory. This beamline uses a double-bounce Si monochrometer and a Ge 111 analyzer to obtain wavevector resolution δq = Å -1 full width at half maximum. Powder samples were in 1 mm diameter glass capillaries in a temperature controlled chamber. 1c - Optical Rotation (OR) We measured the OR by measuring the transmitted intensity for each wavelength 450 nm < λ < 800 nm as a function of the position of the analyzer relative to the polarizer (Figure S8). For an optically inactive, isotropic material, the minimum in transmitted light would occur when the analyzer is orthogonal to the polarizer. We measured the minimum in transmitted intensity for left and right handed domains (Figure S8B); the measured OR is half of the difference between the angles. The dispersion in OR for W508 and GDa226 (Figure S8C) are all well fit by λ λ * φ 2 2 *2, λ λ λ 2-2 -

4 derived by assuming that the optical activity is produced by a chiral modulation of the dielectric tensor on a length-scale much smaller than the wavelength of light and takes into consideration the increase in the molecular birefringence as one approaches the UV absorption bands (S4). The molecular absorptions fits are qualitatively consistent with the measured absorption peaks in dichloromethane: λ * = 310 nm for W508 and λ * = 276 nm for GDa226. In CITRO the OR measurement was not carried out because of the stray birefringence of the B7 domains. 2 - NOTES 2a - Relation to Pasteur s Experiment Pasteur demonstrated that isolated molecules can exist in stable enantiomeric (mirror image) forms by noticing that aqueous solution-grown crystals of sodium ammonium tartarate exhibited two distinct crystal shapes related by mirror reflection (S5). By physically separating and redissolving the two types of crystals he obtained two solutions showing opposite optical rotation, i.e. opposite chirality, demonstrating that even when separated the molecules were handed and structured at least to the extent that they had a permanent memory of their handedness. This beautiful experiment (S6,S7) was based on the fact that in the particular crystallization process used by Pasteur the enantiomers were immiscible, spontaneously phase separating ("deracemizing") into homochiral crystals, each made up of either left- or right-handed molecules. On the other hand, dissolving the unseparated enantiomers produced no optical rotation, and thus an optically achiral or "racemic" solution, indicating that the enantiomers were homogeneously mixed and not phase-separated in solution. This spontaneous appearance of macroscopic chiral domains, currently referred to as conglomerate formation, is now a commonly observed feature of crystals of both chiral and achiral organic molecules. Nevertheless, conglomerate formation is delicate and difficult to predict - even Pasteur's sodium ammonium tartarate yields optically achiral crystals of half left- and half-right handed molecules if the crystals are grown a little above room temperature. Conglomerate formation requires specific and sufficiently strong intermolecular interactions to expel molecules of the wrong handedness. Such interactions are possible in the packed periodic environments of crystals, but clearly absent in the dilute disordered environment of a solution leading to the question of how disordered a phase can be and still spontaneously separate left from right. The only conglomerates known were crystals until 1997 when the B2 layered phases of achiral bent-core molecules (Figure 1) were shown to exhibit macroscopic chiral domains (S8). These phases are fluid smectics, having a single broken translational symmetry in the form of the smectic layering, showing that systems with fluid degrees of freedom - 3 -

5 can propagate homochirality. The dark conglomerates extend the observation of macrosopic spontaneous chirality to isotropic liquids having only short-ranged smectic order. 2b - Electro-Optics The DC phases observed exhibit a wide variety of electric field effects. In many DC forming materials, including the ones discussed here, application of a sufficiently strong electric field transforms the DC texture to the birefringent texture of the ordered B2 phases. This can be understood in terms of a field-induced torque on the polarization and resulting reorientation of the previously randomly ordered layers. In many materials, the DC texture coexists with either the B2 or B7 phases. Of the materials studied here, GDa226 is the only material which entirely fills a display cell with the DC texture when cooled from the high temperature isotropic (HiTIso) phase. Application of field E > 3.5 V/m induces an irreversible transition to a chiral antiferroelectric SmC A P A B2 texture. W508 and CITRO cool from the HiTIso to form coexisting DC and B7 domains, which yield SmC A P A (W508) and SmC S P F (CITRO) B2 phases upon field application. Several other electro-optical effects have been observed and remain to be explained. For some materials, the transition between the field-off DC texture and a birefringent texture under application of an electric field was shown to be reversible, leading to a proposal for the use of DCs in displays (S9). On the other hand, the DC texture has appeared upon field removal from a birefringent phase or when cooling with an applied field (S10). The field application may serve to switch the local layer structure into one consistent with layer curvature (Figure S9), thereby promoting DC formation. In addition, the chirality of the optical activity in the DC domains was observed to flip upon switching the sign of the field (S11). 2c - Electro-Optics of Layer Stackings Consistent with Curvature Layer curvature as described in the text limits the layer stackings to those where the layer tilt in the top (or bottom) halves of the layers is coplanar for all layers, requiring chiral bilayer stackings or racemic four-layer stackings of layers (Figure S9). The clock phases presented here are consistent with the electro-optical observations of Refs. S13 and S14. In the field-off state, the birefringence would be relatively low, and the optic axis along the layer normal. With sufficiently strong field, the simplest electro-optical response is for the molecules to rotate about the layer normal while maintaining their chirality, resulting in the SmC A P F. The SmC A P F again would have optic axis along the layer normal, but would have a different birefringence than the - 4 -

6 field-off state. Thus, one would expect ferroelectric switching, and both the field-on and field-off states to have SmA-like optical behavior, but with different birefringences. Though these structures are chiral (Figure S9), they are formed by an equal mixture of left- and right-handed layers, and thus may account for the observed DC regions of low optical activity. 2d - Focal Conic Defects as a Model for the Dark Conglomerate Layer Structure In a bulk smectic not treated in any particular way to align the layers, the condition of constant layer spacing results in arrays of focal conic domains as the dominant mode of layer organization. Focal conics are structures of curved nested layers, described mathematically by the cyclides of Dupin: families of surfaces of constant spacing having discontinuities only along lines which are ideally confocal conic sections in orthogonal planes. Their ubiquity in smectics is an indication that focal conics represent the least energetically costly way to introduce disorder into a bulk fluid smectic, and thus may be the natural response of a layering system to internal strain. Focal conics are classified as FC I or FC II defects, depending respectively on whether the sign of the Gaussian curvature of their layers is negative or positive (S19). In fact, the images of Figure 2 are dominated by layer curvature motifs already very familiar in FC I textures, displaying a disordered continuum of focal conic-like structures, with consistent appearance of regions having negative Gaussian curvature. Figure 3A shows for example the line-circle focal conic-like domain in W508 in Figure 2A. Thus, motivated by the freeze fracture TEM images (2A,B, 3A), we propose that an appropriate model for the layer structure in the disordered DC layering system is an FC I domain. Figure 3C shows that this structural unit can tile space with very little contribution from FC IIs. As described below, we calculated the free energy of such a volume and found it to be minimized for domain size, a ~ 40d ~200 nm, consistent with the observed structures. Note that R p d, and so the tendency for DC formation should grow stronger in homologous series as d is reduced, a correlation that is indeed observed (S12)

7 2e - Elastic Modeling of Dark Conglomerate Layer Deformation Saddle Splay Relieving Compressional Stress - 2D Model - We wish to consider a two dimensional model of the situation considered in the main text (Figure 4F). In particular we derive the elastic energy of two sheets, compressed in orthogonal directions and then allowed to curve into a saddle shaped deformation. We find that a saddle shaped distortion of this system can help to relieve the compressional stress - stabilizing, for instance, a toroidal focal conic defect relative to flat layers. We consider two isotropic elastic sheets - each sheet is isotropic in the sheet and compressed in this plane by β, resulting in an expansion in the orthogonal direction of 1/β. A similar derivation of an anisotropic media yields a similar result. We imagine that the sheets are then joined with orthogonal compressional directions. As the sheets are very thin relative to their extension, we will follow the Landau and Lifshitz derivation for thin plates (S15). We will derive the displacements due to both A thin plate is distorted from flat so that the neutral (midplane, z = 0) has a displacement of u z (z = 0) = ζ(x, y) = (σ x 2 + σ y 2 )/2. For a saddle shaped deformation, σ > 0 and σ < 0 (or vice versa). The principle radii of curvature of the layer at the origin are R = 1/σ,R = 1/σ. is The free energy of an isotropic medium as a function of the strain tensor (u ij = 1 2 ( ui + u ) j j i ) ( F = µ u ik 1 ) 2 3 δ iku ll Mu2 ll. (1) The components of the stress tensor in terms of the strain tensor are: σ ik = Mu ll δ ik + 2µ (u ik 1 ) 3 δ iku ll. (2) We choose a coordinate system with the origin at the layer midplane, which corresponds to the neutral surface. Assuming the plate is thin, then the x and y components of the displacement at the midplane are negligible, and so u x (z = 0) = u y (z = 0) = 0. In addition, the stresses,σ xz, σ yz, and σ zz are zero on the top and bottom of the plate, and, for a thin plate, must be zero everywhere: σ xz = σ yz = σ zz = 0. Using these relations, we can solve for the displacements. First, from σ xz = 0 and σ yz = 0 compression and bending, and express the free energy of the system in terms of these displacements. -6-

8 we can derive the displacements in x and y. For example: Integrating gives u x z = uz x ζ(x,y) x = σ x. u x σ xz, u y σ yz. (3) Second, u z is found by considering: σ zz = 0 = M(u xx + u yy + u zz ) µ(2u zz (u xx + u yy )). Rearranging gives u zz = u z z u z z = 2µ 3M 4µ + 3M (u yy + u xx ) (4) = 2µ 3M 4µ + 3M (σ + σ )z (5) so u z (z) = ζ(x, y) z2 (σ +σ ) 2µ 3M ζ(x, y) since z is alway small. 2 4µ+3M Now that we have the displacements for the saddle deformation, we need the displacement vectors for the original compression. For the top layer x = (1+β)x so u x = x x = βx, and y = y/(1 + β), so that x y = xy, or the transformation is area preserving. The magnitude of β is determined from the molecular tilt, but is always relatively small. For instance, if the molecule is tilted by 30, then β 0.1. To first order in β, u y = (1/(1 + β) 1)y = β/(1 + β)y βy. Thus by equation 4, u z z 2µ 3M (β β) = 0. (6) 4µ + 3M Thus to lowest order, u z is constant, by symmetry, u z 0. To lowest order, we can just consider these two displacements as adding, so that u x = βx σ xz, u y = βy σ yz u z = (σ x 2 + σ y 2 )/2. (7) The nonzero components of the strain tensor are u xx = β σ z, u yy = β σ z. (8) The free energy is given in terms of the non-zero components of the strain tensor: F = µ ( u ik 1δ ) 2 3 iku ll Mu2 ll (9) 2 = µ ( ) u 2 3 xx u xx u yy + u 2 yy + 1 M(u 2 xx + u yy ) 2 (10) -7-

9 = 2µ(β 2 + βz(σ σ ) σ σ z 2 ) + ( 2 3 µ M)z2 (σ + σ ) 2. (11) Integrating over the half thickness, and including the bottom layer (a factor of two), the free energy per unit volume for a single layer of thickness of d is f bilayer 2 d/2 = d 0 F dz (12) f bilayer = 2µβ 2 + ( 2 3 µ M)d2 12 (σ + σ ) 2 d2 6 µσ σ + µβ d 2 (σ σ ) (13) = f 0 + K 1 2 (σ + σ ) 2 + Kσ σ + G(σ σ ) (14) where K = ( 2 3 µ M)d2 12, K = d 2 6 µ, G = µβ d 2, and f 0 = 2µβ 2 is independent of the radii of curvature. We can identify the terms as a constant, the mean curvature, the Gaussian curvature and an additional term which we will call F twist due to the mismatch between tilt direction in adjacent layers. If the surface is minimal, (σ = σ = σ ), then F bilayer = K(σ Ḡ K )2 + f 0, revealing R = K = d as the preferred radius of curvature. Thus, the G 3β preferred radius of curvature is set by the layer thickness and the tilt in the two half layers. The molecular tilt determines the initial compression of the half sheets, or β. Relation to the Free Energy of the SmC* - Up to this point the elastic model is not chiral. However, among common smectic liquid crystalline phases, only the chiral SmC* phase is consistent with the symmetry of the elastic model. For example, restraining the model to the C symmetry of the SmA phase removes the anisotropic elastic strain. A SmAlike phase made of twisted H shaped molecules, so that there is 2D nematic order in the top and bottom halves of the layer, is however, consistent with the model (S16). For the SmC phase, the c director must lie orthogonal to one of the two C2 symmetry axes in the model which are diagonal to the two principle curvature directions. The addition of the c director breaks the mirror symmetry of the model, yielding the SmC*. In particular, in the case presented here, once the elastic sheets are dressed with their respective molecular half-arms, the structure becomes distinctly chiral to the extent that these tilt directions are rigorously maintained by each molecule in a given region of layer. The elastic model is characterized by the stretch of the upper and lower elastic sheets in orthogonal directions, the tilt directions of the half-molecular arms. The principal curvatures induced in the composite layer are in these directions. Thus it is the tilt of the half molecular arms that are along the directions of principal layer curvature. The layer is made by connecting the molecular half-arms across the layer midplane, which renders the structure chiral. With the choice of molcular arm tilts indicated in Figure 4, c, the vector giving the tilt direction of the molecular plane (Figure -8-

10 1A) points up and right on the page (c = a z 1 a z 2) and the polarization p pointing opposite the arrow of the molecular bow. That is the c and p are mutually orthogonal and at 45 to the molecular arm tilt direction and the principal curvature directions. The term G(σ σ ) driving the saddle splay curvature can be written in general in terms of c and p as p L c, where L is the layer curvature tensor, first identified for tilted chiral smectics by degennes (S17). This term is one of four terms present in the free energy of chiral smectic C phases that are not present in achiral ones (S17). They are F chiral = D 1 (c c p) D 2 (p L c) + D 3 (n c p) + D 4 (p ζ) (15) where n is the layer normal, c is the projection of the tilt direction onto the layer plane, p is the polarization direction, and ζ is the dilation of the layers. The first term is polarization splay, seen in the B7 phase. The third term promotes the helical winding of the c director, as seen in calamitic SmC*s and the B3 phase. The last term involves compression of the layers. Finally, the term of interest here is the second term, which we call F twist because it promotes twist of the layer along the polarization direction. If the polar direction makes an angle α with one of the principal directions of curvature and c is orthogonal to the polarization direction (consistent with C2 symmetry), then in the coordinate system defined by the principal directions of curvature the second term is F twist = D 2 (cos(α), sin(α)) σ 0 0 σ sin(α) cos(α) (16) where the choice of sign of the right hand term is given by the chirality of the phase. As a result, F twist = D 2 (σ σ )sin(2α)/2. Note that it is easier to see how this term does not depend on the choice of n than in our construction above: if n n, then c c, and so the sign of F twist remains unchanged (recall that L is proportional to derivatives of n). Interestingly, unlike the Gaussian curvature, this term is not a total derivative and so does not depend solely on the surface topology (S18). The addition of this term to lipid bilayers has been considered by Helfrich and Prost (S19). In particular they showed that it stabilized the formation of helical ribbons and tubes. Toric Focal Conic Defect - As described in the text, the conical core region of a TFCD is a representative volume of the DC structure. A TFCD is constructed by introducing -9-

11 layers whose normal lie along the lines connecting a circle (x = acos(u), y = asin(u)) and a line (z = asinh(v)). Let r be the distance from the circle along such a connecting line. The distance to the line z = asinh(v) is acosh(v) r, and these two distances are the two principle radii of curvature: σ = 1/r, σ = 1/(acosh(v) r) (they have opposite sign). A volume element is given by a2 σ σ (σ σ ) 2 du dv dr. The mean and Gaussian energy contributions of focal conic defects have been derived by Kléman and Lavrentovich(S20). We consider free energy contributions of mean curvature, Gaussian curvature, and the twist term defined above. F = 1 2 K(σ + σ ) 2 + Kσ σ + G(σ σ ) (17) where K > 0 and G > 0, but K < 0 to penalize pure saddle splay. Care must be taken that σ < σ over the entire integration volume. Note that even though it may be an important contribution to the free energy, we don t include polarization splay, as that requires solving for the polarization direction on the whole surface. The third term is particular easy to integrate: F twist = G Rewriting in terms of r, u, and v gives (σ σ ) a2 σ σ (σ σ ) 2du dv dr. (18) F twist a 2 σ σ = G σ σ 2π = G 0 sinh 1 (1) sinh 1 ( 1) du dv dr (19) acosh(v) 0 a dr dv du (20) cosh(v) = G4πa 2 sinh 1 (1). (21) The sinh 1 (1) factor limits the volume we consider to a cone with radius and height both a. The total volume is V = 2π 3 a3. The mean and Gaussian energy terms have been calculated by Kléman and Lavrentovich (S20) in an infinite volume, we use those values here, as the curvature is greatest near the core of the defect. Using their energies, the total free energy density is F 3π a 2 (K(ln a r c + ln 2 2) K) 6 a Gsinh 1 (1)). (22) -10-

12 Ignoring the slowly varying ln a r c dependence, the free energy is minimized if a = a min = π(k(ln a r c + ln 2 2) K) Gsinh 1 (1) (23) The structure is stable relative to flat layers if a > a min /2. If K = η K then a min = π( K)(η(ln a r c + ln 2 2) + 1) Gsinh 1 (1)) = πr(η(ln a r c + ln2 2) + 1). (24) sinh 1 (1)) If we assume that a 100r c in the logarithmic term, then a min 10 d (3η + 1) (25) which is very consistent with the observed structure, assuming that η

13 3 - FIGURES S1-S9 Figure S1 I(q) q (Å -1 ) Figure S1: First, second and third harmonic diffuse reflections from the layering in GDa226 at T = 100 ºC. The width of the red line indicates the instrumental resolution width, δq = Å -1. The peaks are broadened beyond the resolution due to the finite extent of the regions of local layer flatness

14 Figure S2 Figure S2: Lamellar lattice vector magnitude and peak full width at half height (FWHH) versus temperature from data like that in Figure S1. The fundamental (n = 1) peak width after correction for the resolution, Δq Å -1, indicates a correlation length ξ 2/Δq 1200Å for layer ordering. ξ values for GDa226, W508, and CITRO are indicated in Figure 1. Peak width increases with the harmonic order n because the loss of layer coherence due to layer curvature in the focal conic domains. The increase roughly as Δq n ~ n is expected for saddle splay layer curvature

15 Figure S3 Figure S3: FFTEM image of a 4 µm x 4 µm fracture area at the W508 / glass interface showing the uniformity and isotropy of the DC texture over micron dimensions

16 Figure S4 Figure S4: Higher magnification of an area of Figure S3. The fingerprint texture of the layers indicated that the fracture plane is parallel to the W508 / glass interface and that the preferred layer orientation at the glass has the layer normal parallel to the glass surface. The line tracks a layer across the image, illustrating the layer continuity in the texture, a key requirement for the long-ranged macroscopic chirality. The purple segments follow the intersection of the layer with the surface and the cyan segments are in the third dimension, along nearly cylindrical surfaces near the focal conic cores

17 Figure S5 Figure S5: W508 texture at yet higher magnification, revealing a distinct bilayer structure in this DC phase. The yellow bar has a length L = 4d, where d is the layer spacing, encompassing two bilayers. This bilayer structure produces a subharmonic in the XRD of the W508 DC, shown in Figure S6. The line tracks a layer across the image, illustrating the layer continuity in the texture, a key requirement for the long-ranged macroscopic chirality. The purple segments follow the intersection of the layer with the surface and the cyan segments are in the third dimension, along nearly cylindrical surfaces near the focal conic cores

18 Figure S6 temperature (ºC) I(q) q (Å -1 ) q (Å -1 ) Figure S6: XRD from W508 showing the fundamental diffuse reflection at q = Å -1 and a subharmonic peak at q = Å -1 indicating a local bilayer structure. The bilayer structure is also visualized in the FFTEM images (Figure S5). The bilayer structure does not appear to be specifically related to DC formation, as W508 is the only DC we have studied with bilayer structure. The fundamental and bilayer peak full width at half-max = Å -1, indicating layer correlations over a 500Å length

19 Figure S7 Figure S7: FFTEM image of the B7 phase of W508 showing nearly flat layers with layer steps and the polarization splay undulations, in contrast to Figure 2E which is strongly focal conic. At the lower left a break in the replica and its shadow indicate the shadowing direction (arrow). The B7 coexists with the DC phase in W508 in the FFTEM cells. In capillaries only the DC phase appears on cooling. The DC phase does not appear to be undulated in W508. The B7 undulations, driven by polarization splay, thus appear to be a mode of response to in-layer frustration that competes with the saddle splay of the DC: the B7 is a not saddle splayed whereas the DC is not polarization splayed. As Figure 2E shows the B7 undulations interact with cylindrical layer curvature, having a strong preference for running normal to the axis of curvature

20 Figure S8 λ λ Figure S8: Dispersion of the optical activity the DC phases. (A) The intensity of transmitted light was measured for a range of decrossing angles of the analyzer relative to the polarizer using a spectrophotometer. For each wavelength, the minimum in transmitted intensity was found by fitting a quadratic to the measured intensity vs. analyzer position curves obtained. The minimum occurs when the decrossing angle is equal to the net optical rotation through the sample. In order to avoid errors in determining the relative angles of the analyzer and polarizer, we averaged the optical rotations determined from left and right handed domains. (B) The optical rotation as a function of wavelength for W508 and GDa226. CITRO could not be measured because of the stray birefringence of the coexisting B7 domains, visible in Figure 2K. The optical rotation increases as the wavelength approaches the UV absorption of the molecules

21 Figure S9 Figure S9: (A) The bent-core molecular geometry favors layer saddle splay curvature only when the tilt directions of the top (and bottom) half molecular arms of adjacent layers are coplanar. For example, in (A) and (B), the tilt in the top half of all of the layers is in the East-West direction, while the bottom half of the layer is tilted in the North-South direction. (B) This criterion is satisfied in the B2 bilayer systems for the chiral SmC S P F and SmC A P A structures in (B), but not for the racemic SmC S P A and SmC A P F bilayer structures, the basic reason why the DCs are, but for one case (S13, S14), chiral. (C,D) Racemic four-layer systems with a clock phaselike 90º rotation of p between the layers also satisfy the above criterion. This clock phase is chiral overall even though the layer chirality alternates between left and right handed in adjacent layers. The polarization, p, and the director, c, both helix about the layer normal, though in opposite directions. Such a racemic stacking may account for the non-optically-active DC phase observed in Refs. S13 and S14, as well as for some of their peculiar electro-optic switching behavior (see NOTE 2b)

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