Structure of Liquid Crystal Droplets with Chiral Propeller Texture

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J. Phys. Chem. B 2008, 112, 1358-1366

Structure of Liquid Crystal Droplets with Chiral Propeller Texture Deng-Ke Yang,*,† Kwang-Un Jeong,‡ and S. Z. D. Cheng§ Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State UniVersity, Kent, Ohio 44242; School of AdVanced Materials Engineering, Chonbuk National UniVersity, Jeonju, 561-756, Korea; and Maurice Morton Institute and Department of Polymer Science, The UniVersity of Akron, Akron, Ohio 44325 ReceiVed: August 21, 2007; In Final Form: NoVember 6, 2007

We experimentally studied a nematic liquid crystal whose molecules form twisted head-to-head H-bonded dimers. We observed that when the material transformed from the isotropic to nematic phase, it formed droplets with chiral propeller textures. We carried out a computer simulation to investigate the liquid crystal director configuration inside the droplets and to study the effects of elastic constants and chirality on the droplet texture. Results of our study show it is likely that the material in the droplets had nonzero chirality due to spontaneous chiral phase separation.

1. Introduction Self-assembled liquid crystal (LC) materials formed through noncovalent interactions have attracted numerous attention due to their applications in the biological and electro-optical science and technology.1-4 Hydrogen (H)-bonding is such an example, which has become a major research topic due to its moderate bonding energy (10-50 kJ/mol), directionality, selectivity, and reversibility.5-12 In particular, chirality may be generated when two achiral molecules form a dimer through H-bonding.13-20 The molecular chirality has a profound effect on the structure and liquid crystal meosphase behavior of the dimers. We experimentally studied the structure and mesophase behavior of an achiral asymmetric R,ω-carboxylic acidhydroxyl terminated mesogenic molecule containing an alkoxyl chain with seven carbons (BPCA-C7-PmOH) whose molecular structure is shown in Figure 1. These molecules form dimers through H-bonding. The formed dimers can be axial chiral conformers and can be either left-handed or right-handed.21-25 The system is a racemic mixture consisting of equal amounts of left- and right-handed dimers and exhibits liquid crystal phases. In-situ FTIR study showed that at temperature below the isotropic-liquid crystal phase transition, the dimers are dynamically stable, namely, they do not switch constantly between the right-handed and left-handed conformations.23 More than 80% of the material are in the dimer state. The material was sandwiched between two parallel unrubbed glass plates and was placed inside a temperature controller. Upon cooling, at 218 °C, some of the material transformed from the isotropic to nematic phase and formed droplets with an untwisted cross texture. The residual material remained in the isotropic phase. When the temperature was decreased further to 216 °C, additional material transformed into a liquid crystal phase and formed droplets with chiral propeller textures together with the untwisted cross texture as shown in Figure 2, whereas the droplets with the untwisted cross remained their texture.21-25 * Author for correspondence. E-mail: [email protected]. † Kent State University. ‡ Chonbuk National University. § The University of Akron.

The droplets were embedded in the isotropic background. When the temperature was decreased to 214 °C, the material transformed into the smectic A phase. One interesting speculation arises that dimers with different handednesses separate to form chiral droplets. The liquid crystal inside the droplets with the chiral propeller texture is not nematic (N) but is chiral nematic (N*) with a nonzero chirality. Chiral nematic droplets have been an interesting subject for many years and have attracted much attention.23-26 The liquid crystal field inside a chiral nematic droplet, and thus the droplet texture, depends on the anchoring condition at the droplet surface. When the anchoring condition is tangential, the droplet exhibits the Frank-Pryce spherulitic texture with concentric rings, which have been extensively studied and well understood.32-34 The distance between two consecutive rings is half pitch. Inside the droplet the liquid crystal director is independent of the azimuthal angle. The crossed dark brushes are along radial directions. The elastic deformation of the liquid crystal director is mainly twist deformation. When the anchoring is perpendicular, the droplet structure and texture are less studied. Furthermore, nematic liquid crystal droplets can also exhibit chiral textures because the total elastic energy may be reduced by introducing twist deformation.34-36 Among the three liquid crystal deformations of bend, splay, and twist, twist deformation costs the least energy. This paper presents a computer simulation study on liquid crystal droplets with a perpendicular anchoring condition. We investigated the effects of chirality and elastic constants. We tried to determine whether the liquid crystal inside the droplets with chiral propeller textures is chiral or not. 2. Simulation Technique We consider a liquid crystal droplet with oblate shape. The z-axis of the Cartesian coordinate is along the short axis of the oblate, which is also perpendicular to the substrate. The radius along the short axis is b, and the radius along the long axis is a. The liquid crystal director configuration inside the droplet is described byn b(x,y,z). On the droplet surface, the liquid crystal director is anchored perpendicular to the surface. This assumption is based on the experimental result. The texture shown in

10.1021/jp076719b CCC: $40.75 © 2008 American Chemical Society Published on Web 01/15/2008

LC Droplets with Chiral Propeller Texture

J. Phys. Chem. B, Vol. 112, No. 5, 2008 1359

Figure 1. Chemical structure of the liquid crystal molecule BPCA-C7-PmOH and its dimer.

Figure 2. Microphotograph of the droplets with left-handed propeller, right-handed propeller, and untwisted cross textures:21 (a) without quarter waveplate; (b) with quarter waveplate.

The total elastic energy is

F)

∫-aa∫-aa∫-bbf dx dy dz

(2)

Because the liquid crystal director varies more than 180° inside the droplet, it is better to use the tensor representation of the liquid crystal director in order to prevent the problem of antiparallel orientation of the liquid crystal director at two neighboring lattice sites in the simulation. In the calculation of the elastic energy we use the tensor defined by refs 38 and 39

1 Q 6 (x,y,z) ) b nb n - 6I 3 Figure 3. Schematic diagram showing how the texture is calculated.

Figure 2 is the microphotograph under crossed polarizer and analyzer. On the boundary, the region where the surface normal is parallel or perpendicular to the polarizer is black. This indicates that the liquid crystal director at the boundary is either parallel or perpendicular to the surface normal. Furthermore, the droplets have chiral propeller texture. Therefore, the liquid crystal director must be perpendicular to the surface. There is no externally applied field. We only need to consider the elastic energy of the deformation of the liquid crystal in determining the liquid crystal director configuration inside the droplet. The elastic energy density is given by ref 37

1 1 1 f ) K11(∇‚n b)2 + K22(n b‚∇ × b n )2 + K33(n b×∇×b n )2 (1) 2 2 2

(3a)

where 6I is the 3 × 3 identity tensor. Under the tensor representation, the elastic energy is given by

1 1 (-K11 + 3K22 + K33)G1 + (K11 - K22)G2 + 12 2 1 1 (-K11 + K33)G6 - q0K22G4 + (-K11 + 3K22 + K33)∇‚ 4 12 (n b∇‚n b+b n×∇×b n )] (3b)

f)

where q0 is the chirality of the liquid crystal, which is related to the pitch P by q0 ) 2π/P. The Gi (i ) 1, 2, 4, 6) are defined by

G1 )

∂Qjk ∂Qjk ∂xl ∂xl

(4a)

G2 )

∂Qjk ∂Qjl ∂xk ∂xl

(4b)

1360 J. Phys. Chem. B, Vol. 112, No. 5, 2008

G4 ) ejklQjm G6 ) Qjk

Yang et al.

∂Qkm ∂xl

(4c)

∂Qlm ∂Qlm ∂xj ∂xk

(4d)

The volume integration of the last term in eq 3b can be converted into a surface integration. This term does not affect the liquid crystal director configuration when the orientation is fixed at the boundary but must be included in order to obtain the accurate total elastic energy. In the simulation, a Mx × My × Mz mesh is imposed on the prism containing the droplet. Mx ) My ) 32 and Mz ) (b/a)Mx. The lattice constant is ∆x ) 2a/Mx. The Gi and thus the elastic energy is numerically calculated using the values of b n at the lattice sites. The liquid crystal director configuration in the equilibrium state is calculated using the over-relaxation method. The variation of the director at step τ + 1 of the iteration is calculated by

Figure 4. Liquid crystal director configuration with the parameters K22 ) 5.0 × 10-12 N and P ) ∞.

1 (-K11 + 3K22 + K33)Hτ1 12 1 1 (K - K22)Hτ2 - (-K11 + K33)Hτ6 + K22q0Hτ4 (5) 2 11 4

[

) R(∆x)2 ∆n(τ+1) i

]

where R is the relaxation constant. The Hi (i ) 1, 2, 3, 4) are defined by

H1 )

H2 ) H4 )

H6 )

δG1 ∂2Qji ) -4nj δni ∂xl ∂xl

(

(6a)

)

δG2 ∂2Qil ∂2Qjl ) -2nj + δni ∂xi ∂xl ∂xj ∂xl

(

)

(6b)

δG4 ∂Qli ∂Qlj ) -2nj ejkl + eikl δni ∂xk ∂xk

(

(6c)

)

δG6 ∂Qkl ∂Qji ∂2Qji ∂Qkl ∂Qkl ) -2nj 2 + 2Qkl (6d) δni ∂xk ∂xl ∂xl∂xk ∂xi ∂xj

Because b n is a unit vector, in updating b n, we use the renormalization b n(τ+1) ) (n bτ + ∆n b(τ+1))/|n bτ + ∆n b(τ+1)|. When the droplet is a sphere with radius R (a ) b ) R) and the nematic liquid crystal (q0 ) 0) is in the radial droplet configuration where the liquid crystal director is oriented along the radial direction of the sphere, there is only splay deformation. In the spherical coordinate with the origin at the center of the droplet, the elastic energy density is

f ) (1/2)K11(∇‚n b)2 ) (1/2)K11(2/r)2

Figure 5. Droplet textures with the parameters K22 ) 5.0 × 10-12 N and P ) ∞: (a) without quarter waveplate; (b) with quarter waveplate.

(7)

The total elastic energy is

∫02πdφ ∫0πsin θ dθ ∫0R[21 K11(2r ) ]r2 dr ) 8πK11R 2

(8)

Figure 6. Liquid crystal director configuration of the unipolar chiral droplet with the parameters K22 ) 3.0 × 10-12 N and P ) ∞.

In the calculation of the light intensity pattern, namely the texture, the 2 × 2 Jones matrix method is used.40,41 The ordinary and extraordinary refractive indices of the liquid crystal are no and b ne, respectively. The droplet is divided into Mz slabs with the thickness ∆h ) 2b/Mz. Within each slab, the direction of the liquid crystal director b n is approximately considered to be uniform. In slab i, the angle between b n and the z-axis is θi, and the angle of the projection of b n on xy plane with respect to the

x-axis is βi. The effective extraordinary refractive index is ne/eff ) neno/[(ne cos θ)2 + (no sin θ)2]1/2. The optical retardation of the slam is Γi ) 2π[ne/eff(z ) i∆h) - no]∆h/λ. The refraction at the droplet surface is small and can be neglected because the refractive index inside the droplet (where the material is in the liquid crystal phase) is close to the refractive index outside the droplet (where the material is in the isotropic phase). After passing the polarizer, the incident light is linearly polarized along

F)



Figure 7. Droplet texture of the unipolar chiral droplet with the parameters K22 ) 3.0 × 10-12 N and P ) ∞: (a) without quarter waveplate; (b) with quarter waveplate.

Figure 8. Liquid crystal director configuration of the bipolar chiral droplet with the parameters K22 ) 3.0 × 10-12 N and P ) ∞.

the x direction and the Jones vector isE Bin ) vector of the outgoing light is calculated by

B Eout(x,y) )

(

)

Eout/x(x,y) ) Eout/y(x,y)

(10).

The Jones

Mz(x,y)

∑ i)1

6 R(βi)‚G 6 (Γi)‚R 6-1(βi)‚E Bin (9)

where

(

cos β - sin β 6 R(βi) ) sin β i cos β i i i G 6 (Γi) )

(

e-iΓi/2 0 0 eiΓi/2

)

)

(10)

(11)

The transmission axis of the analyzer is along the y-axis. The intensity of the outgoing light is

I(x,y) ) |Eout/y(x,y)|2

(12)

In the simulation of the droplet texture, the flat spectrum from 400 to 700 nm is used. The refractive indices used are no ) 1.50 and ne ) 1.53, with which the generated texture resembles the experimentally observed textures. The possible reason for

Figure 9. Droplet texture of the bipolar chiral droplet with the parameters K22 ) 3.0 × 10-12 N and P ) ∞: (a) without quarter waveplate; (b) with quarter waveplate.

the small birefringence is because that the microphotographs were taken at a temperature very close to the nematic-istropic transition. The simulated droplet texture is the light intensity pattern which would be observed under an optical microscope with corssed polarizers. White represents high light intensity while black represents low light intensity. In the simulation of the droplet texture with a quarter waveplate, the waveplate is a uniaxial optical retardation film with the optical retardation 2π∆nd/λo ) π/2, where λo ) 550 nm. The slow (uniaxial) axis is at 45° with respect to the x-axis. The optical retardation (∆nd) is assumed to be wavelengthindependent. 3. Results and Discussion In our simulation, we observe three features of the liquid crystal director inside the droplets. First, the directror configuration always has rotational symmetry around one diameter. The z-axis is chosen parallel to the symmetry axis. Second, if the droplet is a sphere, the symmetry axis can be along any diameter determined by the initial condition. If the droplet is oblate, the symmetry axis is along the short axis of the oblate due to the lower elastic energy. In our experimental study, the droplets were usually oblate (diameter in the substrate plate is larger than the cell thickness), and therefore the symmetry axis is perpendicular to the substrate of the cell. Third, there are two stable director configurations, depending on the initial condition. One of the configurations is asymmetric about the equator plane perpendicular to the symmtry axis, obtained under z-independent double twisted initial states. We name this asymmetric configuration unipolar chiral droplet. The other configuration is approximately antireflection-symmetric about the equator plane, obtained under the initial condition where the liquid crystal director has an opposite azimuthal component above and under the equator plane. This configuration is also usually obtained with an isotropic initial condition. We name this antireflection-symmetric configuration bipolar chiral droplet.


Figure 10. Total elastic energy of the droplet as a function of twist elastic constant.

Yang et al.

Figure 13. Liquid crystal director configuration of the bipolar chiral droplet with the pitch P ) 20 µm.

Figure 11. Liquid crystal director configuration of the unipolar chiral droplet with the pitch P ) 20 µm.

Figure 14. Droplet texture of the bipolar chiral droplet with the pitch P ) 20 µm: (a) without quarter waveplate; (b) with quarter waveplate.

Figure 12. Droplet texture of the unipolar chiral droplet with the pitch P ) 20 µm: (a) without quarter waveplate; (b) with quarter waveplate.

Details of these two configurations will be discussed in more detail. Effect of Elastic Constants. In a nematic radial droplet, splay is the only deformation of the liquid crystal director. The elastic energy density, given by eq 7, diverges when the radius of the coordinate approaches 0. The divergence of the free energy density can be avoided by introducing an isotropic core or introducing a bend deformation.35-37 The isotropic core occurs only in a small droplet with submicron size and is not considered in this paper where only droplets with micron size are studied. Furthermore, among the three elastic deformations of liquid crystal director, the twist deformation usually costs the least

energy. Press and Arrot had shown that, by introducing twist deformation, the splay and bend deformation were reduced and thus the total free energy was reduced.35 The droplets studied by Press and Arrott were hemispheres. They used approximations and derived an analytic solution. From the liquid crystal director configuration, they qualitatively described the texture. Here we study droplets whose shapes are close to a sphere, and we quantatively calculate the texture. We first study the effects of elastic constants. The liquid crystal is a nematic liquid crystal. The pitch is P ) ∞ and the chirality is qo ) 2π/P ) 0. The liquid crystal director configuration inside the droplet depends on the ratios between the three elastic constants. When the elastic constants are K11 ) 6.4 × 10-12 N, K22 ) 5.0 × 10-12 N, and K33 ) 10 × 10-12 N,42 and the droplet is a sphere with the radius R ) 5 µm (a ) b ) R), the director configuration is shown in Figure 4. The xz plane is the vertical plane containing the x-axis and the symmetry axis. The xy plane is the equator plane perpendicular to the symmetry axis. There is no escape to twist deformation because the twist elastic constant is not sufficiently small. There is only one stable configuration, independent of the initial condition. The numerically calculated total elastic energy is 4.11






Figure 19. Polarization azimuthal and ellipticity anges vs position: (a) unipolar chiral droplet; (b) bipolar chiral droplet.


× (4/3)πK33R ) 8.6πK11R, which is slightly higher than the analytically obtained total elastic energy of the radial droplet because of the finite mesh size used. The simulated texture is an untwisted cross as shown in Figure 5. In the figures showing the droplet textures, B P represents the direction of the polarizer, B A represents the direction of the analyzer, and N B0 represents the direction of the quarter waveplate. The droplet R radius is fixed at 5 µm in our simulation study. When the twist elastic constant is decreased to K22 ) 3.0 × 10-12 N while other parameters are unchanged, the director

configuration escapes to the twist deformation. Now there are two stable configurations. The director field in the unipolar chiral droplet configuration is shown in Figure 6. The twist deformation can be seen from the liquid crystal director field on both the xz and xy planes. There is more twisting in the bottom half of the droplet than in the top half; therefore, we call it a unipolar chiral droplet. The total elastic energy is 4.12 × (4/3)πK33R. The corresponding droplet texture is a chiral propeller texture as shown in Figure 7a. The degree of twisting of the dark brushes is small. The handedness of the propeller can be either left-handed or right-handed, depending on the initial condition. The director field in the bipolar chiral droplet configuration is shown in Figure 8. There is also twist deformation as can been seen from the liquid crystal director field on both the xz and xy planes. The director field in the top and bottom halves of the droplet is approximately antireflection-symmetric; therefore, we call it a bipolar chiral droplet. The total elastic energy is 4.05 × (4/3)πK33R, which is slightly lower than that of the unipolar


Figure 20. Droplet texture of the unipolar and bipolar chiral droplets with the pitch P ) 10 µm and birefringence ∆n ) 0.06: (a) unipolar chiral droplet; (b) bipolar chiral droplet.


chiral droplet and is also lower than the numerically calculated elastic energy of the radial droplet. The corresponding droplet textures are shown in Figure 9. Although the director field shows clearly that there is twist deformation, surprisingly the texture is an untwisted cross. After investigating how the polarization of light evolves inside the droplet, we find that in this case the polarization of light rotates first counterclockwise and then clockwise. After passing through the droplet, the ellipticity and direction of the polarization only change slightly. We will discuss this issue in more detail later. When the elastic constant is decreased further, more twisting of the liquid crystal director is introduced, and the propeller texture is twisted more. For most liquid crystal, the ratio between the twist and bend elastic constants is about 3:10 or larger. At this ratio, although the unipolar chiral droplet shows chiral propeller texture, the degree of twisting of the propeller is smaller than that observed in our experiment. We searched for the critical value of the twist elastic constant under which the twist deformation occurs. We calculated the total elastic energy of the bipolar droplet as a function of the twist elastic K22, and the result is plotted in Figure 10. For K22 higher than K22/C ) 5.0 × 10-12 N, the total elastic energy does not change with K22, which indicates that there is no twist

Yang et al.



deformation. For K22 lower than K22/C, the total elastic energy decreases with K22, which means that twist deformation occurs. The decrease of the total elastic energy is faster than a linear relation with K22 because the volume within which twist deformation occurs also increases. Effect of Chirality. Chirality has profound effects on the droplet texture. The chirality q0 of a chiral nematic liquid crystal is related to its pitch P by the relation q0 ) 2π/P. In this study, the elastic constants are K11 ) 6.4 × 10-12 N, K22 ) 5.0 × 10-12 N, and K33 ) 10 × 10-12 N, and the droplet is a sphere with the radius R ) 5 µm (a ) b ) R). For nonzero chirality, there are always two stable configurations: unipolar and bipolar chiral droplets. When the pitch is P ) 20 µm, the director configuration of the unipolar chiral droplet is shown in Figure 11, and the corresponding textures are shown in Figure 12. The total elastic energy is 4.44 × (4/3)πK33R. The director configuration of the bipolar chiral droplet is shown in Figure 13, and the corresponding textures are shown in Figure 14. The total elastic energy is 4.20 × (4/3)πK33R. Although there is twist of the director on the horizontal and vertical planes, the texture is an untwisted cross. As the pitch decreases, the liquid crystal director twists more and thus the propeller of the droplet texture twists more. When the pitch is P ) 10 µm, the director configuration of the unipolar


J. Phys. Chem. B, Vol. 112, No. 5, 2008 1365 4. Discussion and Conclusion


chiral droplet is shown in Figure 15, and the corresponding textures are shown in Figure 16. The total elastic energy is5.17 × (4/3)πK33R. The director configuration of the bipolar chiral droplet is shown in Figure 17, and the corresponding textures are shown in Figure 18. The total elastic energy is 4.69 × (4/ 3)πK33R. For the droplets with the pitch P ) 10 µm, the evolution of the polarization of the light is studied, and the results are shown in Figure 19. The light is incident at the location (x ) -R/2, y ) 0). Inside the droplets, the linearly polarized incident light changes into elliptical polarized light. φ is the azimuthal angle of the major axis of the polarization ellipse with respect to the x-axis. ν is the ellipticity angle. In the unipolar chiral droplet, after the light passing through the droplet, its azimuthal and ellipticity angles change significantly, and the spot is white on the texture (the dark brush twists away). In the bipolar chiral droplet, after light passing through the droplet, its azimuthal and ellipticity angles only change slightly, and the spot is black on the texture (the dark brush does not twist from that point). We also studied the effects of birefringence. When the birefringence is ∆n ) 0.06, the simulated textures are shown in Figure 20. When the birefringence is increased, the dark brushes of the unipolar droplet texture become narrow, while the bipolar droplet texture is still untwisted cross. When the pitch is decreased to 5 µm, the liquid crystal director configuration of the unipolar chiral droplet is shown in Figure 21, and the corresponding textures are shown in Figure 22. The total elastic energy is 7.84 × (4/3)πK33R. The liquid crystal director configuration of the bipolar chiral droplet is shown in Figure 23, and the corresponding textures are shown in Figure 24. The total elastic energy is 7.08 × (4/3)πK33R. Now the pitch is the same as the radius, and the director configuration becomes complex. The liquid crystal director twists more than 360° inside the droplet, and the droplet textures have a black ring in the middle.

The simulation study shows that nematic liquid crystal droplets with perpendicular anchoring may exhibit chiral propeller texture, agreeing with analytical studies.34-36 When the twist elastic is sufficiently smaller than the splay and bend elastic constants, the total elastic energy can be reduced by escaping from the splay and bend deformation into the twist deformation. The simulation also shows that chiral droplets with twist deformation may not exhibit chiral propeller texture. There are two possible director configurations inside the chiral droplets. One of the configurations is the bipolar chiral droplet where the liquid crystal director is antireflection-symmetric about the equator plane in the sense that the director has the opposite azimuthal components about the equator plane. The other configuration is the unipolar chiral droplet where there is no antireflection symmetry. Although the unipolar droplet configuration has a higher elastic energy, it can still exist because its elastic energy is a local minimum. The initial condition determines the configuration of the droplet. Both the bipolar and unipolar chiral droplets have twist deformation of the director on the equator plane. The twist deformation does not necessary generate the chiral propeller texture because the director varies with the z coordinate (which is perpendicular to the equator plane). The texture of the droplets is determined by the overall optical effect of the liquid crystal. The unipolar chiral droplet generates the chiral propeller texture while the bipolar chiral droplet does not. These two different director configurations and the corresponding textures occur independent of whether the chiral director configuration is produced by the intrinsic chirality of the liquid crystal or by the twist deformation escape. Qualitatively speaking of the experimentally observed droplet textures, the droplets could be either chiral droplets generated by intrinsic chirality or by twist deformation escape. Under the hypothesis of twist deformation escape, the droplets with the right-handed and left-handed chiral propeller textures have the unipolar chiral droplet configuration, and the droplet with untwisted cross texture has the bipolar chiral droplet configuration. The twist deformation escape can produce both lefthanded and right-handed chiral propeller textures. Under the hypothesis of intrinsic chirality, the droplets with the righthanded and left-handed chiral propeller textures have the unipolar chiral droplet configuration, and the droplet with untwisted cross texture has the bipolar chiral droplet configuration or nematic radial droplet configuration. The intrinsic chirality is caused by spontaneous chiral phase separation which was reported in smectic liquid crystal systems by Takezoe et al.43,44 The droplet with left-handed chiral propeller texture has a left-handed chirality while the droplet with right-handed chiral propeller texture has a right-handed chirality. The droplet with the untwisted cross texture may have the chirality of zero, namely, no spontaneous chiral phase separation. The formation of the droplets provides the required confinement for the spontaneous chiral phase separation. Our simulation does not consider the intermolecular interaction energy difference between molecules with different handedness and thus cannot predict the spontaneous chiral phase separation. If the experimentally observed droplet textures are quantitatively compared with the simulated droplet textures, we tend to conclude that the chiral propeller textures are caused by spontaneous chiral phase separation. Under the hypothesis of twist deformation escape, in order to generate the degree of the twist of the experimentally observed chiral propeller texture, the twist elastic constant K22 has to be as small as 1.0 × 10-12

1366 J. Phys. Chem. B, Vol. 112, No. 5, 2008 N (when the bend elastic constant K33 is 10.0 × 10-12 N). This value of the twist elastic constant is unrealistically small. Unfortunately, it is very difficult to experimentally measure the elastic constants because the material has a very small dielectric anisotropy and does not readily reorient under electric fields. Under the hypothesis of the spontaneous chiral phase separation, the liquid crystal in the droplets has the intrinsic pitch about 10 µm. Acknowledgment. We thank Professor Oleg Lavrentovich for very useful discussions. This work was partially supported by NSF (DMR-0516602). References and Notes (1) Goodby, J. W. In Handbook of Liquid Crystals; Demus, D., Goodby, J., Gray, G. W., Spiess, H.-W., Vill, V., Eds.; Wiley-VCH: Weinheim, 1998; Vol. 1, pp 115-132. (2) Hirschberg, J. H. K. K.; Brunsveld, L.; Ramzi, A.; Vekemans, J. A. J. M.; Sijbesma, R. P.; Meijer, E. W. Nature (London) 2000, 407, 167. (3) Mizoshita, N.; Kanie, K. Macromol. Rapid Commun. 2001, 22, 797. (4) Reinhoudt, D. N.; Crego-Calama, M. Science 2002, 295, 2403. (5) Yang, W.; Chai, X.; Chi, L.; Liu, X.; Cao, Y.; Lu, R.; Jiang, Y.; Tang, X.; Fuchs, H.; Li, T. Chem.sEur. J. 1999, 5, 1144. (6) Hirschberg, J. H.; Brunsveld, L.; Ramzi, A.; Vekemans, J. A.; Sijbesma, R. P.; Meijer, E. W. Nature (London) 2000, 407, 167. (7) Kato, T.; Mizoshita, N.; Kanie, K. Macromol. Rapid Commun. 2001, 22, 797. (8) Kato, T. Science 2002, 295, 2414. (9) Reinhoudt, D. N.; Drego-Calama, M. Science 2002, 295, 2403. (10) Keinan, S.; Ratner, M. A.; Marks, T. J. Chem. Mater. 2004, 16, 1848. (11) Xue, C.; Jin, S.; Weng, X.; Ge, J. J.; Shen, Z.; Shen, H.; Graham, M. J.; Jeong, K.-U.; Wang, H.; Zhang, D.; Guo, M.; Harris, F. W.; Cheng, S. Z. D.; Li, C. Y.; Zhu, L. Chem. Mater. 2004, 16, 1014. (12) Jin, S.; Ma, Y.; Zimmerman, S. C.; Cheng, S. Z. D. Chem. Mater. 2004, 16, 2975. (13) Mason, S. F. Nature (London) 1983, 311, 19. (14) Goodby, J. W. Science 1986, 231, 350. (15) Li, C. Y.; Cheng, S. Z. D.; Ge, J. J.; Bai, F.; Zhang, J. Z.; Mann, I. K.; Chien, L. C.; Harris, F. W.; Lotz, B. J. Am. Chem. Soc. 2000, 122, 72. (16) Li, C. Y.; Cheng, S. Z. D.; Weng, X.; Ge, J. J.; Bai, F.; Zhang, J. Z.; Calhoun, B. H.; Harris, F. W.; Chien, L. C.; Lotz, B. J. Am. Chem. Soc. 2000, 123, 2462.

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Structure of Liquid Crystal Droplets with Chiral Propeller Texture

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