Spatially periodic liquid crystal director field appearing in a photonic crystal template pectroscopy

Spatially periodic liquid crystal director field appearing

in a photonic crystal template

Heinrich Matthias, Thorsten Röder, Ralf B. Wehrspohn, and Heinz-S. Kitzerowa兲 Department of Chemistry, Faculty of Science, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Sven Matthias

Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany Stephen J. Picken

Delft University of Technology, Dept. Material Science and Technology, Julianalaan 136, Delft 2628 BL, The Netherlands

共Received 15 March 2005; accepted 12 October 2005; published online 6 December 2005兲 Active tuning of photonic crystals can be achieved by filling the porous structures with liquid crystals. Here, the director field in macropores was studied by fluorescence confocal polarizing microscopy. For this purpose, the photonic crystal was infiltrated with a glass-forming liquid crystalline polymer, the sample was cooled below the glass transition temperature and, subsequently, the photonic crystal template was removed. Results on a structure with modulated pores indicate a spatially periodic director field containing a lattice of disclination rings. © 2005 American Institute

of Physics. 关DOI:10.1063/1.2142100兴

Liquid crystals are liquids with anisotropic optical, elec-trical, magnetical, and mechanical properties which are very sensitive to temperature and external fields. The change of the degree of order with temperature and the alignment of the director n in external fields can be used to change the difference of the principle refractive indices or the orienta-tion of the optical axis, respectively. Consequently, the pho-tonic properties of a phopho-tonic crystal containing a liquid crystal can be actively modulated.1–4A suitable model sys-tem is macroporous silicon,5,6where regular arrays of pores with an extremely high aspect ratio 共depth : diameter 艋100␮m : 1␮m兲 can be fabricated by electrochemical etching. Here, experimental and theoretical studies of the di-rector fields of liquid crystals within such pores are de-scribed. For liquid crystals in cylindrical cavities with uni-form diameter d⬎0.1␮m, an escaped radial director field is known to be stable.7,8 Here, we report for the first time on microscopic studies of the director field in pores with a spa-tially periodic diameter variation.

The director fields and the optical properties are studied experimentally and are calculated numerically. One major class of algorithms for the latter purpose starts from the Frank–Oseen vector representation of the free energy den-sity, which was transformed by Dickmann to an alignment tensor representation preserving the equivalence of n and −n.9,10The equilibrium state of the director is characterized by the minimum free energy, obtained by integrating the free energy density over the volume and applying the Euler– Lagrange method. In the absence of electric or magnetic fields and chirality, the Euler–Lagrange equation reads 0 = −关fs兴a_jkwith the functional derivative

10 关fs兴a_jk= 2

冉

1 12K11− 1 4K22− 1 12K33

冊

ajk,ll−共K11− K22 − K33兲ajl,lk+ K24ajl,lk+ 1 4共K33− K11兲共alm,jalm,k − a_lm,la_jk,m− almajk,ml− alm,majk,l− almajk,lm兲. 共1兲 The alignment tensor a is defined by ajk= 1 / 2 共3njnk−␦jk兲, a_jk,l=⳵ajk/⳵xl, and ajk,lm=⳵2ajk/共⳵xl⳵xm兲. The scalar order parameter, S, is assumed to be independent on the position, as surface interactions are expected to affect S only in pores with a diameter below 100 nm.11,12 The elastic coefficients

K11, K22, and K33 describe the elastic energy due to splay, twist, and bend deformations, respectively. The constant K₂₄ corresponds to a mixed term which is relevant only for samples with large surface/volume ratio.

Here, the dynamic equation of the director ␥1共⳵ajk/⳵t兲 = −关fs兴a_jk is solved numerically, where ␥1 is the rotational viscosity of the liquid crystal. Discretizing time, this leads to the following algorithm:

a␶+1jk = ajk␶ − ⌬t

␥1

关fs兴a_jk. 共2兲

After each iteration step, the director was normalized to en-sure that the alignment tensor remains symmetric and trace-less. In the simplifying case of the one-constant approxima-tion 共K₁₁= K₂₂= K₃₃, K₂₄= 0兲, this approach is equivalent to the well-known algorithm of Kilian and Hess.13

For our experimental studies, we used a three-dimensional orthorhombic photonic crystal consisting of modulated macropores in silicon.5,6The pores are ordered in the hexagonal lattice with a lattice constant of 4.2␮m and the pore diameter was modulated with depth showing a modulation length of 11␮m. The pore diameter was varying sinusoidally between 2.2 and 3.3␮m共Fig. 1兲. After cleaning in an ultrasonic bath and with a plasma cleaner, the silicon wafer was pretreated with DMOAP to support strong

homeo-a兲_{Electronic mail: kitzerow@chemie.upb.de}

APPLIED PHYSICS LETTERS 87, 241105共2005兲

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tropic anchoring.14 Subsequently, we filled the pores in a vacuum with the nematic liquid crystal polymer ASY 10 关Fig. 2共a兲兴, which shows a glasslike nematic state 共g兲 in the temperature range below the nematic共N兲 phase. The phase transition temperatures of ASY 10 are: g 46 ° C N 137 ° C

Iso. For fluorescence polarizing microscopy, this compound

was doped with N , N

⬘

-bis共2,5-di-tert-butylphenyl兲-3,4,9,10-perylene-carboximide共BTBP兲 关Fig. 2共b兲兴. After an-nealing in the nematic phase at 120 ° C for 24 h, the macroporous structures were cooled to room temperature, thereby freezing the director in the glassy state. Subse-quently, the silicon wafer was dissolved in concentrated aqueous KOH solution. The remaining isolated polymer rods were washed and investigated by fluorescence confocal po-larizing microscopy共FCPM兲.15

Like standard fluorescence confocal microscopy, FCPM uses a scanning laser beam focused on the sample. But in addition, the sample is doped with a small amount 共⬍0.1 wt %兲 of an anisometric fluorescent dye 共BTBP, in our case兲. The transition dipole moment of the dichroic dye is oriented along the local director of the liquid crystal host.16 The incident laser beam共488 nm,Ar+_{兲 and the} emit-ted light pass a polarizer, which implies that the intensity of the detected light scales as I⬀ cos4␣for an angle␣between the local director and the electric-field vector of the polarized light. As the fluorescence wavelength of BTBP peaks at ⬇540 nm, the exciting and the fluorescence light can be ef-fectively separated by a filter with an absorption edge at 510 nm. One well-known problem arising from applying FCPM on liquid crystals is the birefringence-induced defocusing16 which can be roughly estimated by d共⌬n/n_av兲, where⌬n is the birefringence, navis the average refractive index, and d is the scanning depth. Our measurements on ASY 10 at 50 ° C 共i.e., slightly above the glass transition temperature兲 indicate ⌬n=0.218 and nav= 1.616. Since the polymer rods are observed in the transverse direction, the maximum value of d is in the order of the pore diameter 共3.3␮m兲, yielding a defocusing in the order of 0.5␮m.

In order to interpret the FCPM pictures, we have written computer programs based on Algorithm共2兲. To this end, the shape of the pores was described by a polynomial, whereas the volume of a pore was discretized by 63⫻63⫻131 grid points of a constant mesh size. The differential terms emerg-ing in Eq.共1兲 were accordingly approximated by central

dif-ferences, e.g., ajk,x⬇共ajk关x+1,y,z兴−ajk关x−1,y,z兴兲/共2⌬x兲. The values of the elastic coefficients of the nematic side chain polymer used were estimated as follows. According to Fabre et al.,17the elastic constants of side chain polymers are of the same order of magnitude as the coefficients of the corresponding monomers. The ratio K33/ K11 increases with decreasing length of the spacer, whereas the degree of poly-merization has only a minor impact on the elastic properties. Accounting for the very short spacer of ASY 10, we made the following estimations: K11= 10 pN, K22= 5 pN, and K33 = 20 pN. Crawford et al.18demonstrated that the saddle splay elastic constant is in the range of the medium elastic con-stant. Thus, we assumed K₂₄= 10 pN. The simulations turned out to show only minor changes in the resulting director fields by using modified elastic constants.

The simulations started from the isotropic phase, charac-terized by a random orientation of the directors. According to the treatment with DMOAP, a fixed homeotropic anchoring is used in the simulations, throughout. After 10 000 iteration steps, the energy inside the simulation volume changed only by an amount smaller than 0.004% per iteration, which we considered as a sign for attaining the equilibrium. Figure 3共a兲 reproduces the director field in the central plane of the pore. Two disclination loops appear with the topological charges

s = ± 1 / 2. The loop of strength +1 / 2 unfolds in the belly of

each pore, whereas the loop of strength −1 / 2 appears in the neck of each pore. This is in accordance with the theory of defects, which predicts the spreading of hyperbolic and hedgehog defects into disclination loops of half-integer strength.19

Based on these director patterns, we simulated the FCPM pictures by integrating the fluorescence intensity over the z resolution of the confocal microscope共0.5␮m兲 starting from the middle of the pore共Fig. 3兲. The fluorescence inten-sity of each molecule was scaled according to I⬀ cos4_{␣, as} described above. In comparison to the pictures measured by FCPM, there is a reasonable conformance.

For a further examination, we took pictures of the rods at an angle of␸⬇45° between the pore axis and the plane of the electric-field vector of the incident light. For a director field with axial symmetry, the parts of the sample, in which the director is tilted toward and away from the plane of po-larization by this rotation, are equal. In agreement with this expectation, an alternating pattern of fluorescence brightness is observed共Fig. 4兲.

In conclusion, both theoretical and FCPM studies of the director fields of modulated macropores reveal an escaped radial director field, in good agreement with each other. In

FIG. 2. Chemical structures of the nematic polymer ASY 10共a兲, and the dichroic dye BTBP共b兲.

FIG. 1. Electron micrograph showing the side view of a three-dimensional macroporous silicon structure.

241105-2 Matthias et al. Appl. Phys. Lett. 87, 241105共2005兲

the cylindrical cavities studied previously, pointlike hedge-hog and hyperbolic defects appear at random positions and tend to disappear after annealing, due to the attractive forces between defects of opposite topological charges. In contrast, the modulated pores used in this study stabilize a periodic array of disclinations. Moreover, disclination loops appear instead of pointlike disclinations. Apart from their interesting topology, these samples may also be valuable for the appli-cation in tunable photonic crystals. Previous studies have shown that nearly identical samples which were not treated

with the surface-active agent DMOAP exhibit a uniform par-allel alignment. An anchoring transition from parpar-allel to per-pendicular anchoring, as observed in nematic droplets,20 would change the effective refractive index for light propa-gating along the pore axis from the ordinary value no to an average value of the ordinary noand the extraordinary refrac-tive index ne, thereby affecting the photonic properties, considerably.

The authors would like to thank the German Research Foundation共DFG兲 for financial support 共SPP No. 1113, and Project Nos. KI 411/5 and WE 2637/6兲.

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strength +1 / 2共waist of the pore兲 and of strength −1/2 共neck of the pore兲. 共b兲–共e兲 Calculated 关共b兲 and 共d兲兴 and experimental 关共c兲 and 共e兲兴 FCPM images of modulated silicon pores with a maximal diameter of 3.3␮m. The plane of polarization is parallel关共b兲 and 共c兲兴 and perpendicular 关共d兲 and 共e兲兴 to the pore axis, respectively. The simulated patterns are obtained by integrating the intensity of light over a distance of 0.5␮m which corresponds to the optical resolution in z direction.

FIG. 4.共a兲 FCPM picture of a liquid crystalline rod. The angle between the plane of polarization and the pore axis is␸⬇42°. 共b兲 Simulated pattern.

241105-3 Matthias et al. Appl. Phys. Lett. 87, 241105共2005兲