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The interaction of colloidal particles with weak homeotropic anchoring energy in
Sung-Jo Kim and Jong-Hyun Kim
Department of Physics, Chungnam National University, 220 Gung-dong, Yuseong-gu, Daejeon 305-764 Korea. E-mail : [email protected]
We have investigated interactions of colloidal particles with weak homeotropic anchoring energy in homogeneous nematic liquid crystal cells. Particle-wall and inter-particle interactions were observed experimentally and analyzed using typical dipole-dipole and quadrupole-quadrupole interactions, including substrate effects as the image charges. Both experimental results matched well with the calculated results for the effective particle radius reflecting the weak anchoring. The effective radius is reduced by the amount of extrapolation length than the actual particle radius. The effective radii of polyethylene micro-particles were reduced to a coefficient ζ (0.78 ≥ ζ ≥ 0.52) times the actual radius with anchoring coefficients in the range of 3.8×10-6 ~ 1.4×10-6 J/m2. The anchoring energy of the particles is, therefore, a key component for explaining liquid crystal colloidal systems.
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homogeneous nematic liquid crystal cells
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1. Introduction
Recently, numerous studies of nematic liquid crystal (NLC) colloids have been performed. The interactions associated with the surface conditions of particles indicate diversity in the nematic bulk. Surfaces that contain
hedgehog or Saturn-ring defects [1–6]. On the other hand, surfaces with strong planar anchoring energies are accompanied by boojum defects [1, 6]. Attractive or repulsive forces between particles accompanied by hyperbolic hedgehog defects are proportional to r-4, and those between particles accompanied by Saturn-ring defects are proportional to r-6. Here, r is the distance between the centers of mass of two particles. In addition, the forces between particles accompanied by boojum defects are proportional to r-6. The forces between particle accompanied by hyperbolic hedgehog defect and particle accompanied by Saturn-ring defect or boojum defect are proportional to r-5 [1–4, 7–12]. Such a particle-defect pair is considered to be a dipole or quadrupole moment, and the forces between the pairs are considered dipole-dipole-like, quadrupolequadrupole-like, or dipole-quadrupole-like interactions. These pairs, which we call defect configurations, are designated dipolar configuration (DC, for particles accompanying a hyperbolic hedgehog defect-radial hedgehog defect pair), and quadrupolar configuration (QC, for particles accompanying a Saturn-ring defectradial hedgehog defect pair) in this report. Studies of non-spherical colloidal particles have been reported variously [13–18]. The directions of the dipole and quadrupole moments of non-spherical particles are affected by their shape, the anchoring energy conditions, and the far-field director orientation (n), while the directions of spherical particles are determined by the anchoring energy condition and n only. Moreover, the ability to form tunable shapes leads to the possibility for various colloidal structures. The inter-particle forces in non-spherical particles are qualitatively similar to that in spherical particles, but the coefficients are different. The surface of a Janus particle is composed of two anchoring conditions. The spherical Janus particle is accompanied by both hyperbolic hedgehog and Saturn-ring defects with strong homeotropic anchoring energy in one hemisphere and strong planar anchoring energy in the other hemisphere [19, 20]. These particles interact as a combined dipole and quadrupole moment. In addition, because both the shape and anchoring condition of the particle can be altered, diverse structures can be derived.
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spherical colloidal particles with strong homeotropic anchoring energy are accompanied by hyperbolic
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The aforementioned inter-particle forces are described for a nematic bulk with strong anchoring energy. In the nematic film, we must consider the effects of substrates. The interactions between the substrate wall and the particles have been studied experimentally and theoretically [2–4, 21–31]. There are three methods for explaining the interactions, namely, the current [4, 20], the coat [4, 22–26], and the mirror-image
quadrupole-wall interactions are proportional to r-4 and r-6, respectively. A spherical colloidal particle of radius R with a smooth surface and strong anchoring energy has dipole or quadrupole moments proportional to R2 or R3, respectively. The interactions of spherical colloidal particles having strong anchoring energy have been widely studied. However, the interactions of particles having weak anchoring energy have been not studied. In this report, we have experimentally studied the particle-wall and inter-particle interactions with weak surface anchoring particle. We showed that the experimental results can be explained by a simple model adopting the method of image and the concept of extrapolation. In the “Results and Discussions” section, we introduce the results step by step, dividing the section into “Particle-wall interactions-Discussions” and “Interparticle interactions”.
2. Experiments
We used 4-cyano-4'-pentylbiphenyl (5CB, from Merck) as the liquid crystal. 5CB undergoes a phase transition from a crystalline to a nematic phase at about 24 oC, and it goes transitions from the nematic to the isotropic phase at about 36 oC. At 35 oC, the effective shear viscosity ηeff is 27 × 10-3 Pa·s [32] and the mass density ρ is 1.01 g/cm3 [33]. The elastic constant K of 5CB is considered as 10-11 N in calculations [4]. We prepared the NLC colloid by mixing 5CB with spherical polyethylene micro particles (from Cospheric). Both clear polyethylene microspheres (CPMS) and grey polyethylene microspheres (GRYPMS) were used. The diameters of CPMS and GRYPMS are in the ranges of 20–27 µm and 27–32 µm and their densities are ~0.96 g/cm3 and ~1.00 g/cm3, respectively. The surfaces of both kinds of particles are hydrophobic. The relative density of particle to the density of 5CB is critical factor positioning the equilibrium
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approaches [27–31]. The coefficients of each method have small differences, but in all cases, dipole-wall and
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height of particle in the cell and the above two types of polyethylene particles show the clear difference in height. We prepared homogeneously planar aligned cells. The substrates with a polyimide coating were rubbed uniformly. The cells were of two types, namely, constant-gap cells and wedge cells. The cell gaps were
the shapes are nearly spherical, and the particles have grooves on the surface. We observed hyperbolic hedgehog and Saturn-ring defects around the particles after inserting the NLC colloid into the cells as in FIG. 1(b), (c), (e) and (f). The far-field director of the particle-defect pairs corresponds to either DC or QC. It is evident that the chain structure consisted of particles exhibiting dipole-dipole and quadrupole-quadrupole interactions [1–6]. Hyperbolic hedgehog and Saturn-ring defects around the particles usually appear to be positioned closer to the particles than that of strong anchoring in our experiment. It may be due to the weak anchoring of the polyethylene particles itself and deformation by the groove influence [34]. To obtain fluorescence images, we prepared a mixture of nematic colloids and a fluorescent dye, N,Nbis(2,5-di-tert-butylphenyl)-3,3,9,10-perylenedicarboximide (BTBP, from Sigma Aldrich) at ~0.7 w%. In the NLC, BTBP molecules were arranged parallel to the director. The polarization of the light radiated from BTBP dye is also parallel to the director [29]. A confocal fluorescence microscope (TCS NT, from Leica) was used to observe the vertical position of particles in the cell. The temperature was adjusted with a temperature controller (TMS 94 and LTS 350m, from Linkam). The configuration of the particles, the particle-wall and inter-particle interaction data were obtained through a polarizing optical microscope (Eclipse E600, from Nikon). The vertical positions, which correspond to z-axis in FIG. 2(a), of particles were measured using an optical microscope in the wedge cells. The particles experience the elastic repulsive forces concurrently from both the upper and bottom substrates in the nematic phase and the buoyancy to the upper direction. When the nematic phase of 5CB changes to the isotropic phase at 36 oC, the elastic repulsive forces disappear and only the buoyancy remains. The particles move until they touch the upper substrate in the isotropic phase. If 5CB returns to the nematic phase, the elastic repulsive force reappears and is added to the buoyancy. Therefore, the particles in contact with the upper substrate are pushed downward and return to the equilibrium position away from the substrate. We measured the height difference in the two phases. Because of optical observation, the
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20–120 µm. In FIG. 1, (a) and (d) are optical images of CPMS and GRYPMS respectively. One can see that
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real distance is related to the optical path length, which depends on the refractive index. Thus, the distance was corrected and the correction depends on the refractive indices, the numerical aperture and the radii of particles. The corrected length was approximately equal to the non-corrected length multiplied by the refractive index of liquid crystal. The horizontal distance between particles was determined using snapshots
center of mass of two particles are checked each snapshot.
3. Results and Discussions
We used the method of image to understand both DC and QC interactions. However, this method is not applied directly in our system because the surface anchoring of spherical polyethylene particle is not strong. To overcome such a problem, we thought a simple model. Let us consider following two cases. In one case that two particles are the same size, the deformation with strong anchoring is larger than that with weak anchoring. In other case that both two particles have strong anchoring, the NLC deformation of large size particle is larger than that of small size one. Therefore, the deformation reduction by weak anchoring can be convertible to diminution of effective particle size. We assume the effective radius, R’, that reflects the weak anchoring. Let R'=ζR. The anchoring is strong at the virtual surface of R' and we can effectively replace the weak anchoring particle of radius R as the strong anchoring particle of radius R' for the explaining the experiment results. The coefficient ζ is related with anchoring energy and extrapolation length and it is in the range of 0 δm. This indicates that the elastic repulsive forces obtained from the measurement are smaller than the elastic repulsive forces obtained from the calculation using the ideal situation of strong anchoring for a given particle size. That is, the deformation of particle with weak anchoring is smaller than strong anchoring. As proposed before, we replace R as R’ in Eq. (1) and Eq. (2). When R is substituted by R’, the elastic repulsive forces were reduced for reduced particle sizes. However, the radius of particle R in buoyancy Fg need not be replaced to determine the effective actual buoyancy. Inserting the appropriate R and R' in Eq. (3) and Eq. (4), the equations can be rewritten as:
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DC and QC using side view scanning of a confocal fluorescence microscope in the cell, as in FIG. 2(b), (c)
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ζ4 ζ4 1 0
(5)
ζ6 ζ6 1 0
(6)
From the above equations, it is clear that δ’c > δc, where δc and δ’c are the position shifts of the non-adopted R
actual radius. To calculate coefficient ζ, we used the linear asymptotic lines of F’D and F’Q in the large h range. The coefficients, the slope and the intercept, are found fitting the data with linear lines. We assign the inceptes, ho and h’o , calculated with δc =0 and δ’c=0, respectively. The symbol δ’c explains for the δm. Using these relations, we can derive the equation ζ = h’0/h0. In this experiment, ζ is 0.78±0.01 (CPMS with DC), 0.57±0.01 (CPMS with QC), 0.73±0.02 (GRYPMS with DC), and 0.62±0.01 (GRYPMS with QC).
B. Inter-particle interactions We checked the validity of the concept of the effective radius using an inter-particle interaction using GRYPMS. The cell has a uniform planar alignment with a constant cell gap. Two particles were assumed to be DCs and QCs interacting with each other. The dipole-dipole (FDD) and quadrupole-quadrupole (FQQ) interactions in bulk NLC can be expressed as [4]: 12π r 4 1 3cos2
(7)
20π r 6 9 90cos2 105cos4
(8)
Where θ is the angle between r and n. When a dipole moment p1 is parallel to p2, the force is negative (attractive). However, when p1 is antiparallel to p2, the force is positive (repulsive). We consider the attractive cases. When the dipole-dipole interactions occurs parallel to the substrate, h is constant. We can assume that the particles move only in the y-direction. In this case, r(t) is equal to y(t), and t is the time in seconds. Because the force FDD is attractive in our experiment, the distance between the DCs is steadily diminishing, as in FIG. 4(a). FIGURE 4(b) shows the speed of the particles as a function of the distance, and the inset shows the relation between time and distance. The inset shows that the force between the DCs is proportional to r-4 in
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and adopted R’, respectively. The prime symbol indicates that the effective radius is adopted in place of the
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the cell, but the value of the coefficient does not match Eq. (7), because in this equation, the force is obtained without considering the effects of walls. The sign of the force FQQ is determined by the angle θ and the force is proportional to r-6. In FIG. 4(c), one can see the increasing distance between QCs. FIGURE 4(d) shows the speed of the particles as a function
the coefficient for QCs does not match Eq. (8). For the inter-particle interactions, therefore, we added the forces induced by the walls using the method of image. P1 and P2 are real defect configurations in the cell, as in FIG. 5. P3 and P4 are the imaginary defect configurations of P1. P2 interacts with all P1, P3 and P4. In contrast, P1 interacts with P2 and its imaginary charges. The equation of motion of a particle in NLC is expressed as 2 r 6πηeff !r" [3]. FR is the force felt by an interacting particle, and ηeff is the viscosity of the NLC. Because of the character of the configuration, the motion depends only on the y components of the forces. The total force F2,y acting on P2 by P1, P3 and P4 is a function of the y components and is expressed as: ,$ %& ⋅ ( %&) ⋅ * %&+ ⋅ * %&, ⋅ *
(9)
F21, F23 and F24 are the forces acting on P2 from P1, P3 and P4 and their distances are r21 , r23 , and r24 respectively. We replaced R with R’ in Eq. (9), similar to the previously described particle-wall interaction. Then, Eq. (9) can be rewritten as F’2,y =ζ4F2,y for DCs and F’2,y =ζ6F2,y for QCs. We have fitted this to the experimental result using the modified function. The calculated coefficient ζ is 0.63±0.01 (DCs) and 0.52±0.01 (QCs) in inter-particle interactions.
C. Discussions There was a coat approach to the particle-wall and inter-particle interactions, in which the particle considered to have strong anchoring [26]. Using the coat approach, we calculated the effective radius and the coefficient ζ is approximately 0.65. This is corresponds to the result of our approach. Therefore, the introduction of the effective radius seems to be a suitable method to explain the interactions of particles with weak anchoring energy.
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of distance, and the inset shows the relation between time and distance. Also in the same manner with DCs,
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We obtained the relationship between the coefficient ζ and the anchoring energy using the extrapolation length. As the virtual surface at the radius R′ has strong anchoring, the extrapolation length (L) of particle is expressed as L ~ R-R′ = (1-ζ)R. Using this relation, we acquired that anchoring energy wp of particles is 3.8×10-6 ≥ wp ≥ 1.4×10-6 J/m2. This weak anchoring energy qualitatively matches that reported for
The LC texture on CPMS and GRYPMS polyethylene particles indicates that the particle surface is non-smooth as shown in FIG. 1. We obtained the characteristic length-scale (1.0±0.2 µm) and the roughness (0.033±0.004 µm (CPMS) and 0.039±0.006 µm (GRYPMS)) with an AFM measurement. We should consider the effect of the non-smooth surface on the anchoring. The equivalent anchoring energy, w, arisen out of 1
/02
deformation on regular grooves with weak homeotropic anchoring is expressed as - 4 . 11/L [36]. Where the wave vector is q=2π/λ and extrapolation length is L=K/we (we is surface anchoring energy) [37]. A is the magnitude and λ is the periodicity of grooves. Even though the surface roughness of particles is not regular as the above regular model system, application of the regular model in our system may not cause the serious variation for the limited error range of the AFM measurement. We treat the roughness as A and the characteristic length-scale as the λ. Assuming we=wp as the first approximation, w becomes about 10-8 J/m2. The equivalent anchoring energy is negligible due to wp≫w and we need not to consider the effect of morphology. All the ζ values obtained in the experiments are in the range of 0.52~0.78. When we consider the variation in the size and shape of particles and position accuracy, it seems that the ζ range indicates rather uniform anchoring properties of the particles. However, we can see characteristic trends. Comparing coefficient ζ between DC and QC for the same kind particles throughout the experiments, ζ values of DCs is always slightly larger than the ζ of QCs. This may indicate that, even in the same kind particles, some particles of relatively strong anchoring have DC and the others of relatively weak anchoring have QC. There was no clear indication related to the particle size. In the experiments of particle-wall and inter-particle interaction using GRYPMS particles, the ζ values of inter-particle interaction are smaller than the value of particle-wall interaction for both DC and QC. The dynamic measurement of the inter-particle interaction may perturb the pure interaction compared to the static measurement of particle-wall interaction.
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amorphous polyethylene [35].
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4. Conclusions
We observed the static and dynamic interactions of particles within NLC cells. The particles had two types of configurations, namely, DC and QC. Both particle-wall and inter-particle interactions were considered. The
substrates with strong anchoring, we introduced the method of image to deal with the effect of walls. We used CPMS and GRYPMS polyethylene particles. The experimental results indicated that NLC deformation is reduced compared to the particles with strong anchoring. We introduced the effective particle radius to explain the decrease of deformation. This decreased effective radius indicates a decreased anchoring strength on the particle surface. The anchoring coefficient is 3.8×10-6 ≥ wp ≥ 1.4×10-6 J/m2, which indicates weak anchoring. The experimental and calculated results indicate that the anchoring energy of the particles is important for explaining the behavior of liquid crystal colloidal systems.
Acknowledgements The study was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010–0023379). S.J.K. thanks Prof. K.J.Y. and Mr. D.H.K of CNU for AFM measurement.
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interactions are analyzed through an analogy with electrostatics. Because the NLC colloid is enclosed by two
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References 1. E. M. Terentjev, Phys. Rev. E, 1995, 51, 1330. 2. P. Poulin, H. Stark, T. C. Lubensky and D. A. Weitz, Science, 1997, 275, 1770. 3. P. Poulin, V. Cabuil and D. A. Weitz, Phys. Rev. Lett., 1997, 79, 4862.
5. Y. Gu and N. L. Abbott, Phys. Rev. Lett., 2000, 85, 4719. 6. M. Tasinkevych, N. M. Silvestre and M. M. Telo da Gama, New J. Phys., 2012, 14, 073030. 7. V. G. Nazarenko, A. B. Nych and B. I. Lev, Phys. Rev. Lett., 2001, 87, 075504. 8. M. Yada, J. Yamamoto and H. Yokoyama, Phys. Rev. Lett., 2004, 92, 185501. 9. Musevic, M. Skarabot, U. Tkalec and S. Zumer, Science, 2006, 313, 954. 10. K. Takahashi, M. Ichikawa and Y. Kimura, Phys. Rev. E, 2008, 77, 020703. 11. M. Skarabot, S. Zumer, U. Tkalec, I. Babic, N. Osterman and I. Musevic, Phys. Rev. E, 2008, 77, 031705. 12. I. Musevic and M. Skarabot, Soft Matter, 2008, 4, 195. 13. U. Tkalec, M. Skarabot and I. Musevic, Soft Matter, 2008, 4, 2402. 14. H. Matthias and H. -S. Kitzerow, Mol. Cryst. Liq. Cryst., 2009, 508, 127. 15. F. R. Hung, Phys. Rev. E, 2009, 79, 021705. 16. C. P. Lapointe, T. G. Mason and I. I. Smalyukh, Science, 2009, 326, 1083. 17. B. Senyuk and I. I. Smalyukh, Soft Matter, 2012, 8, 8729. 18. E. Terentjev, Nature Materials, 2013, 12, 187. 19. M. Melle, S. Schlotthauer, M. Mazza, S. H. L. Klapp and M. Schoen, J. Chem. Phys., 2012, 136, 194703. 20. M. Conradi, M. Ravnik, M. Bele, M. Zorko, S. Zumer and I. Musevic, Soft Matter, 2009, 5, 3905. 21. S. B. Chernyshuk and B. I. Lev, Phys. Rev. E, 2010, 81, 041701. 22. B. I. Lev, S. B. Chernyshuk, P. M. Tomchuk and H. Yokoyama, Phys. Rev. E, 2002, 65, 021709. 23. J. Fukuda, B. I. Lev, K. M. Aoki and H. Yokoyama, Phys. Rev. E, 2002, 66, 051711. 24. J. Fukuda, B. I. Lev and H. Yokoyama, Condens. Matter. Phys., 2003, 15, 3841. 25. J. Fukuda and S. Zumer, Phys. Rev. E, 2009, 79, 041703. 26. S. B. Chernyshuk and B. I. Lev, Phys. Rev. E, 2011, 84, 011707. 27. V. M. Pergamenshchik and V. O. Uzunova, Eur. Phys. J. E, 2007, 23, 161.
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4. T. C. Lubensky, D. Pettey and N. Currier, Phys. Rev. E, 1998, 57, 610.
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28. V. M. Pergamenshchik and V. O. Uzunova, Phys. Rev. E, 2007, 76, 011707. 29. O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii and O. D. Lavrentovich, Phys. Rev. Lett., 2007, 99, 127802. 30. V. M. Pergamenshchik and V.O. Uzunova, Phys. Rev. E, 2009, 79, 021704.
32. A. G. Chmielewshi, Mol. Cryst. Liq. Cryst., 1986, 132, 339. 33. G. A. Oweimreen, A. K. Shihab, K. Halhouli and S. F. Sikander, Mol. Cryst. Liq. Cryst., 1986, 138, 327. 34. H. Yokoyama and H. A. van Sprang, J. Appl. Phys., 1985, 57, 4520. 35. T. P. Doerr and P. L. Taylor, Int. J. Mod. Phys. C., 1999, 10, 415. 36. G. Barbero, A. S. Gliozzi and M. Scalerandi, J. Appl. Phys., 2008, 104, 094903. 37. P.G. de Gennes and J. Prost, The physics of liquid crystals (Oxford University press, Oxford New York, 1993)
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31. V.M. Pergamenshchik and V.O. Uzunova, Phys. Rev. E, 2011, 83, 021701.
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Figure Captions
FIGURE 1. (a) and (d) Optical images of CPMS and GRAMS respectively in air. (b), (c), (e) and (f) The
texture of GRAMS. (f) is QC texture of GRAMS. The second images of (b), (c), (e) and (f) are obtained with the rotation of the first images by 45 degree. The white line in the image is indicating the director orientation. The particles in the images are different. The size of images is 40x40 µm2.
FIGURE 2. (a) Schematic diagram of the forces acting on a particle. The substrate surface is under the condition of strong planar anchoring energy along the director n. Forces from both the upper and bottom walls and buoyancy act on the particle. (b), (c) and (d) Side views of the particles in the cell. (c) and (d) are particle images of DC and QC respectively.
FIGURE 3. Calculated δc and measured δm position shifts of the particles. (a) CPMS with DC. (b) CPMS with QC. (c) GRYPMS with DC. (d) GRYPMS with QC. The small inset of each figure shows an optical microscope image of the corresponding defect configuration. The indices c and m are position shift calculated by R and h and measured by an optical microscope respectively.
FIGURE 4. (a) and (c) Movement of two DCs and two QCs in the cell as the function of time. (b) and (d) Time vs. distance and distance vs. speed graphs of dipole-dipole interactions and quadrupole-quadrupole interactions.
FIGURE 5. Schematic diagram of the forces acting on a DC. The dotted circles are the effective radius of a particle.
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polarizing optical images in nematic phase. (b) is DC texture of CPMS. (c) is QC texture of CPMS. (e) is DC
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Figure 2
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Figure 4
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Figure 3
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Figure 5
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The novelty of the work The interactions of weak homeotropic anchoring particles were observed experimentally and analyzed with introducing the image charges and effective radius.
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Colour graphic
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The interaction of colloidal particles with weak homeotropic anchoring energy in
Sung-Jo Kim and Jong-Hyun Kim
Department of Physics, Chungnam National University, 220 Gung-dong, Yuseong-gu, Daejeon 305-764 Korea. E-mail : [email protected]
We have investigated interactions of colloidal particles with weak homeotropic anchoring energy in homogeneous nematic liquid crystal cells. Particle-wall and inter-particle interactions were observed experimentally and analyzed using typical dipole-dipole and quadrupole-quadrupole interactions, including substrate effects as the image charges. Both experimental results matched well with the calculated results for the effective particle radius reflecting the weak anchoring. The effective radius is reduced by the amount of extrapolation length than the actual particle radius. The effective radii of polyethylene micro-particles were reduced to a coefficient ζ (0.78 ≥ ζ ≥ 0.52) times the actual radius with anchoring coefficients in the range of 3.8×10-6 ~ 1.4×10-6 J/m2. The anchoring energy of the particles is, therefore, a key component for explaining liquid crystal colloidal systems.
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homogeneous nematic liquid crystal cells
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1. Introduction
Recently, numerous studies of nematic liquid crystal (NLC) colloids have been performed. The interactions associated with the surface conditions of particles indicate diversity in the nematic bulk. Surfaces that contain
hedgehog or Saturn-ring defects [1–6]. On the other hand, surfaces with strong planar anchoring energies are accompanied by boojum defects [1, 6]. Attractive or repulsive forces between particles accompanied by hyperbolic hedgehog defects are proportional to r-4, and those between particles accompanied by Saturn-ring defects are proportional to r-6. Here, r is the distance between the centers of mass of two particles. In addition, the forces between particles accompanied by boojum defects are proportional to r-6. The forces between particle accompanied by hyperbolic hedgehog defect and particle accompanied by Saturn-ring defect or boojum defect are proportional to r-5 [1–4, 7–12]. Such a particle-defect pair is considered to be a dipole or quadrupole moment, and the forces between the pairs are considered dipole-dipole-like, quadrupolequadrupole-like, or dipole-quadrupole-like interactions. These pairs, which we call defect configurations, are designated dipolar configuration (DC, for particles accompanying a hyperbolic hedgehog defect-radial hedgehog defect pair), and quadrupolar configuration (QC, for particles accompanying a Saturn-ring defectradial hedgehog defect pair) in this report. Studies of non-spherical colloidal particles have been reported variously [13–18]. The directions of the dipole and quadrupole moments of non-spherical particles are affected by their shape, the anchoring energy conditions, and the far-field director orientation (n), while the directions of spherical particles are determined by the anchoring energy condition and n only. Moreover, the ability to form tunable shapes leads to the possibility for various colloidal structures. The inter-particle forces in non-spherical particles are qualitatively similar to that in spherical particles, but the coefficients are different. The surface of a Janus particle is composed of two anchoring conditions. The spherical Janus particle is accompanied by both hyperbolic hedgehog and Saturn-ring defects with strong homeotropic anchoring energy in one hemisphere and strong planar anchoring energy in the other hemisphere [19, 20]. These particles interact as a combined dipole and quadrupole moment. In addition, because both the shape and anchoring condition of the particle can be altered, diverse structures can be derived.
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spherical colloidal particles with strong homeotropic anchoring energy are accompanied by hyperbolic
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The aforementioned inter-particle forces are described for a nematic bulk with strong anchoring energy. In the nematic film, we must consider the effects of substrates. The interactions between the substrate wall and the particles have been studied experimentally and theoretically [2–4, 21–31]. There are three methods for explaining the interactions, namely, the current [4, 20], the coat [4, 22–26], and the mirror-image
quadrupole-wall interactions are proportional to r-4 and r-6, respectively. A spherical colloidal particle of radius R with a smooth surface and strong anchoring energy has dipole or quadrupole moments proportional to R2 or R3, respectively. The interactions of spherical colloidal particles having strong anchoring energy have been widely studied. However, the interactions of particles having weak anchoring energy have been not studied. In this report, we have experimentally studied the particle-wall and inter-particle interactions with weak surface anchoring particle. We showed that the experimental results can be explained by a simple model adopting the method of image and the concept of extrapolation. In the “Results and Discussions” section, we introduce the results step by step, dividing the section into “Particle-wall interactions-Discussions” and “Interparticle interactions”.
2. Experiments
We used 4-cyano-4'-pentylbiphenyl (5CB, from Merck) as the liquid crystal. 5CB undergoes a phase transition from a crystalline to a nematic phase at about 24 oC, and it goes transitions from the nematic to the isotropic phase at about 36 oC. At 35 oC, the effective shear viscosity ηeff is 27 × 10-3 Pa·s [32] and the mass density ρ is 1.01 g/cm3 [33]. The elastic constant K of 5CB is considered as 10-11 N in calculations [4]. We prepared the NLC colloid by mixing 5CB with spherical polyethylene micro particles (from Cospheric). Both clear polyethylene microspheres (CPMS) and grey polyethylene microspheres (GRYPMS) were used. The diameters of CPMS and GRYPMS are in the ranges of 20–27 µm and 27–32 µm and their densities are ~0.96 g/cm3 and ~1.00 g/cm3, respectively. The surfaces of both kinds of particles are hydrophobic. The relative density of particle to the density of 5CB is critical factor positioning the equilibrium
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approaches [27–31]. The coefficients of each method have small differences, but in all cases, dipole-wall and
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height of particle in the cell and the above two types of polyethylene particles show the clear difference in height. We prepared homogeneously planar aligned cells. The substrates with a polyimide coating were rubbed uniformly. The cells were of two types, namely, constant-gap cells and wedge cells. The cell gaps were
the shapes are nearly spherical, and the particles have grooves on the surface. We observed hyperbolic hedgehog and Saturn-ring defects around the particles after inserting the NLC colloid into the cells as in FIG. 1(b), (c), (e) and (f). The far-field director of the particle-defect pairs corresponds to either DC or QC. It is evident that the chain structure consisted of particles exhibiting dipole-dipole and quadrupole-quadrupole interactions [1–6]. Hyperbolic hedgehog and Saturn-ring defects around the particles usually appear to be positioned closer to the particles than that of strong anchoring in our experiment. It may be due to the weak anchoring of the polyethylene particles itself and deformation by the groove influence [34]. To obtain fluorescence images, we prepared a mixture of nematic colloids and a fluorescent dye, N,Nbis(2,5-di-tert-butylphenyl)-3,3,9,10-perylenedicarboximide (BTBP, from Sigma Aldrich) at ~0.7 w%. In the NLC, BTBP molecules were arranged parallel to the director. The polarization of the light radiated from BTBP dye is also parallel to the director [29]. A confocal fluorescence microscope (TCS NT, from Leica) was used to observe the vertical position of particles in the cell. The temperature was adjusted with a temperature controller (TMS 94 and LTS 350m, from Linkam). The configuration of the particles, the particle-wall and inter-particle interaction data were obtained through a polarizing optical microscope (Eclipse E600, from Nikon). The vertical positions, which correspond to z-axis in FIG. 2(a), of particles were measured using an optical microscope in the wedge cells. The particles experience the elastic repulsive forces concurrently from both the upper and bottom substrates in the nematic phase and the buoyancy to the upper direction. When the nematic phase of 5CB changes to the isotropic phase at 36 oC, the elastic repulsive forces disappear and only the buoyancy remains. The particles move until they touch the upper substrate in the isotropic phase. If 5CB returns to the nematic phase, the elastic repulsive force reappears and is added to the buoyancy. Therefore, the particles in contact with the upper substrate are pushed downward and return to the equilibrium position away from the substrate. We measured the height difference in the two phases. Because of optical observation, the
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20–120 µm. In FIG. 1, (a) and (d) are optical images of CPMS and GRYPMS respectively. One can see that
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real distance is related to the optical path length, which depends on the refractive index. Thus, the distance was corrected and the correction depends on the refractive indices, the numerical aperture and the radii of particles. The corrected length was approximately equal to the non-corrected length multiplied by the refractive index of liquid crystal. The horizontal distance between particles was determined using snapshots
center of mass of two particles are checked each snapshot.
3. Results and Discussions
We used the method of image to understand both DC and QC interactions. However, this method is not applied directly in our system because the surface anchoring of spherical polyethylene particle is not strong. To overcome such a problem, we thought a simple model. Let us consider following two cases. In one case that two particles are the same size, the deformation with strong anchoring is larger than that with weak anchoring. In other case that both two particles have strong anchoring, the NLC deformation of large size particle is larger than that of small size one. Therefore, the deformation reduction by weak anchoring can be convertible to diminution of effective particle size. We assume the effective radius, R’, that reflects the weak anchoring. Let R'=ζR. The anchoring is strong at the virtual surface of R' and we can effectively replace the weak anchoring particle of radius R as the strong anchoring particle of radius R' for the explaining the experiment results. The coefficient ζ is related with anchoring energy and extrapolation length and it is in the range of 0 δm. This indicates that the elastic repulsive forces obtained from the measurement are smaller than the elastic repulsive forces obtained from the calculation using the ideal situation of strong anchoring for a given particle size. That is, the deformation of particle with weak anchoring is smaller than strong anchoring. As proposed before, we replace R as R’ in Eq. (1) and Eq. (2). When R is substituted by R’, the elastic repulsive forces were reduced for reduced particle sizes. However, the radius of particle R in buoyancy Fg need not be replaced to determine the effective actual buoyancy. Inserting the appropriate R and R' in Eq. (3) and Eq. (4), the equations can be rewritten as:
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DC and QC using side view scanning of a confocal fluorescence microscope in the cell, as in FIG. 2(b), (c)
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ζ4 ζ4 1 0
(5)
ζ6 ζ6 1 0
(6)
From the above equations, it is clear that δ’c > δc, where δc and δ’c are the position shifts of the non-adopted R
actual radius. To calculate coefficient ζ, we used the linear asymptotic lines of F’D and F’Q in the large h range. The coefficients, the slope and the intercept, are found fitting the data with linear lines. We assign the inceptes, ho and h’o , calculated with δc =0 and δ’c=0, respectively. The symbol δ’c explains for the δm. Using these relations, we can derive the equation ζ = h’0/h0. In this experiment, ζ is 0.78±0.01 (CPMS with DC), 0.57±0.01 (CPMS with QC), 0.73±0.02 (GRYPMS with DC), and 0.62±0.01 (GRYPMS with QC).
B. Inter-particle interactions We checked the validity of the concept of the effective radius using an inter-particle interaction using GRYPMS. The cell has a uniform planar alignment with a constant cell gap. Two particles were assumed to be DCs and QCs interacting with each other. The dipole-dipole (FDD) and quadrupole-quadrupole (FQQ) interactions in bulk NLC can be expressed as [4]: 12π r 4 1 3cos2
(7)
20π r 6 9 90cos2 105cos4
(8)
Where θ is the angle between r and n. When a dipole moment p1 is parallel to p2, the force is negative (attractive). However, when p1 is antiparallel to p2, the force is positive (repulsive). We consider the attractive cases. When the dipole-dipole interactions occurs parallel to the substrate, h is constant. We can assume that the particles move only in the y-direction. In this case, r(t) is equal to y(t), and t is the time in seconds. Because the force FDD is attractive in our experiment, the distance between the DCs is steadily diminishing, as in FIG. 4(a). FIGURE 4(b) shows the speed of the particles as a function of the distance, and the inset shows the relation between time and distance. The inset shows that the force between the DCs is proportional to r-4 in
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and adopted R’, respectively. The prime symbol indicates that the effective radius is adopted in place of the
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the cell, but the value of the coefficient does not match Eq. (7), because in this equation, the force is obtained without considering the effects of walls. The sign of the force FQQ is determined by the angle θ and the force is proportional to r-6. In FIG. 4(c), one can see the increasing distance between QCs. FIGURE 4(d) shows the speed of the particles as a function
the coefficient for QCs does not match Eq. (8). For the inter-particle interactions, therefore, we added the forces induced by the walls using the method of image. P1 and P2 are real defect configurations in the cell, as in FIG. 5. P3 and P4 are the imaginary defect configurations of P1. P2 interacts with all P1, P3 and P4. In contrast, P1 interacts with P2 and its imaginary charges. The equation of motion of a particle in NLC is expressed as 2 r 6πηeff !r" [3]. FR is the force felt by an interacting particle, and ηeff is the viscosity of the NLC. Because of the character of the configuration, the motion depends only on the y components of the forces. The total force F2,y acting on P2 by P1, P3 and P4 is a function of the y components and is expressed as: ,$ %& ⋅ ( %&) ⋅ * %&+ ⋅ * %&, ⋅ *
(9)
F21, F23 and F24 are the forces acting on P2 from P1, P3 and P4 and their distances are r21 , r23 , and r24 respectively. We replaced R with R’ in Eq. (9), similar to the previously described particle-wall interaction. Then, Eq. (9) can be rewritten as F’2,y =ζ4F2,y for DCs and F’2,y =ζ6F2,y for QCs. We have fitted this to the experimental result using the modified function. The calculated coefficient ζ is 0.63±0.01 (DCs) and 0.52±0.01 (QCs) in inter-particle interactions.
C. Discussions There was a coat approach to the particle-wall and inter-particle interactions, in which the particle considered to have strong anchoring [26]. Using the coat approach, we calculated the effective radius and the coefficient ζ is approximately 0.65. This is corresponds to the result of our approach. Therefore, the introduction of the effective radius seems to be a suitable method to explain the interactions of particles with weak anchoring energy.
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of distance, and the inset shows the relation between time and distance. Also in the same manner with DCs,
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We obtained the relationship between the coefficient ζ and the anchoring energy using the extrapolation length. As the virtual surface at the radius R′ has strong anchoring, the extrapolation length (L) of particle is expressed as L ~ R-R′ = (1-ζ)R. Using this relation, we acquired that anchoring energy wp of particles is 3.8×10-6 ≥ wp ≥ 1.4×10-6 J/m2. This weak anchoring energy qualitatively matches that reported for
The LC texture on CPMS and GRYPMS polyethylene particles indicates that the particle surface is non-smooth as shown in FIG. 1. We obtained the characteristic length-scale (1.0±0.2 µm) and the roughness (0.033±0.004 µm (CPMS) and 0.039±0.006 µm (GRYPMS)) with an AFM measurement. We should consider the effect of the non-smooth surface on the anchoring. The equivalent anchoring energy, w, arisen out of 1
/02
deformation on regular grooves with weak homeotropic anchoring is expressed as - 4 . 11/L [36]. Where the wave vector is q=2π/λ and extrapolation length is L=K/we (we is surface anchoring energy) [37]. A is the magnitude and λ is the periodicity of grooves. Even though the surface roughness of particles is not regular as the above regular model system, application of the regular model in our system may not cause the serious variation for the limited error range of the AFM measurement. We treat the roughness as A and the characteristic length-scale as the λ. Assuming we=wp as the first approximation, w becomes about 10-8 J/m2. The equivalent anchoring energy is negligible due to wp≫w and we need not to consider the effect of morphology. All the ζ values obtained in the experiments are in the range of 0.52~0.78. When we consider the variation in the size and shape of particles and position accuracy, it seems that the ζ range indicates rather uniform anchoring properties of the particles. However, we can see characteristic trends. Comparing coefficient ζ between DC and QC for the same kind particles throughout the experiments, ζ values of DCs is always slightly larger than the ζ of QCs. This may indicate that, even in the same kind particles, some particles of relatively strong anchoring have DC and the others of relatively weak anchoring have QC. There was no clear indication related to the particle size. In the experiments of particle-wall and inter-particle interaction using GRYPMS particles, the ζ values of inter-particle interaction are smaller than the value of particle-wall interaction for both DC and QC. The dynamic measurement of the inter-particle interaction may perturb the pure interaction compared to the static measurement of particle-wall interaction.
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amorphous polyethylene [35].
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4. Conclusions
We observed the static and dynamic interactions of particles within NLC cells. The particles had two types of configurations, namely, DC and QC. Both particle-wall and inter-particle interactions were considered. The
substrates with strong anchoring, we introduced the method of image to deal with the effect of walls. We used CPMS and GRYPMS polyethylene particles. The experimental results indicated that NLC deformation is reduced compared to the particles with strong anchoring. We introduced the effective particle radius to explain the decrease of deformation. This decreased effective radius indicates a decreased anchoring strength on the particle surface. The anchoring coefficient is 3.8×10-6 ≥ wp ≥ 1.4×10-6 J/m2, which indicates weak anchoring. The experimental and calculated results indicate that the anchoring energy of the particles is important for explaining the behavior of liquid crystal colloidal systems.
Acknowledgements The study was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010–0023379). S.J.K. thanks Prof. K.J.Y. and Mr. D.H.K of CNU for AFM measurement.
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interactions are analyzed through an analogy with electrostatics. Because the NLC colloid is enclosed by two
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References 1. E. M. Terentjev, Phys. Rev. E, 1995, 51, 1330. 2. P. Poulin, H. Stark, T. C. Lubensky and D. A. Weitz, Science, 1997, 275, 1770. 3. P. Poulin, V. Cabuil and D. A. Weitz, Phys. Rev. Lett., 1997, 79, 4862.
5. Y. Gu and N. L. Abbott, Phys. Rev. Lett., 2000, 85, 4719. 6. M. Tasinkevych, N. M. Silvestre and M. M. Telo da Gama, New J. Phys., 2012, 14, 073030. 7. V. G. Nazarenko, A. B. Nych and B. I. Lev, Phys. Rev. Lett., 2001, 87, 075504. 8. M. Yada, J. Yamamoto and H. Yokoyama, Phys. Rev. Lett., 2004, 92, 185501. 9. Musevic, M. Skarabot, U. Tkalec and S. Zumer, Science, 2006, 313, 954. 10. K. Takahashi, M. Ichikawa and Y. Kimura, Phys. Rev. E, 2008, 77, 020703. 11. M. Skarabot, S. Zumer, U. Tkalec, I. Babic, N. Osterman and I. Musevic, Phys. Rev. E, 2008, 77, 031705. 12. I. Musevic and M. Skarabot, Soft Matter, 2008, 4, 195. 13. U. Tkalec, M. Skarabot and I. Musevic, Soft Matter, 2008, 4, 2402. 14. H. Matthias and H. -S. Kitzerow, Mol. Cryst. Liq. Cryst., 2009, 508, 127. 15. F. R. Hung, Phys. Rev. E, 2009, 79, 021705. 16. C. P. Lapointe, T. G. Mason and I. I. Smalyukh, Science, 2009, 326, 1083. 17. B. Senyuk and I. I. Smalyukh, Soft Matter, 2012, 8, 8729. 18. E. Terentjev, Nature Materials, 2013, 12, 187. 19. M. Melle, S. Schlotthauer, M. Mazza, S. H. L. Klapp and M. Schoen, J. Chem. Phys., 2012, 136, 194703. 20. M. Conradi, M. Ravnik, M. Bele, M. Zorko, S. Zumer and I. Musevic, Soft Matter, 2009, 5, 3905. 21. S. B. Chernyshuk and B. I. Lev, Phys. Rev. E, 2010, 81, 041701. 22. B. I. Lev, S. B. Chernyshuk, P. M. Tomchuk and H. Yokoyama, Phys. Rev. E, 2002, 65, 021709. 23. J. Fukuda, B. I. Lev, K. M. Aoki and H. Yokoyama, Phys. Rev. E, 2002, 66, 051711. 24. J. Fukuda, B. I. Lev and H. Yokoyama, Condens. Matter. Phys., 2003, 15, 3841. 25. J. Fukuda and S. Zumer, Phys. Rev. E, 2009, 79, 041703. 26. S. B. Chernyshuk and B. I. Lev, Phys. Rev. E, 2011, 84, 011707. 27. V. M. Pergamenshchik and V. O. Uzunova, Eur. Phys. J. E, 2007, 23, 161.
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4. T. C. Lubensky, D. Pettey and N. Currier, Phys. Rev. E, 1998, 57, 610.
Page 13 of 18
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DOI: 10.1039/C3SM53092B
28. V. M. Pergamenshchik and V. O. Uzunova, Phys. Rev. E, 2007, 76, 011707. 29. O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii and O. D. Lavrentovich, Phys. Rev. Lett., 2007, 99, 127802. 30. V. M. Pergamenshchik and V.O. Uzunova, Phys. Rev. E, 2009, 79, 021704.
32. A. G. Chmielewshi, Mol. Cryst. Liq. Cryst., 1986, 132, 339. 33. G. A. Oweimreen, A. K. Shihab, K. Halhouli and S. F. Sikander, Mol. Cryst. Liq. Cryst., 1986, 138, 327. 34. H. Yokoyama and H. A. van Sprang, J. Appl. Phys., 1985, 57, 4520. 35. T. P. Doerr and P. L. Taylor, Int. J. Mod. Phys. C., 1999, 10, 415. 36. G. Barbero, A. S. Gliozzi and M. Scalerandi, J. Appl. Phys., 2008, 104, 094903. 37. P.G. de Gennes and J. Prost, The physics of liquid crystals (Oxford University press, Oxford New York, 1993)
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31. V.M. Pergamenshchik and V.O. Uzunova, Phys. Rev. E, 2011, 83, 021701.
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Figure Captions
FIGURE 1. (a) and (d) Optical images of CPMS and GRAMS respectively in air. (b), (c), (e) and (f) The
texture of GRAMS. (f) is QC texture of GRAMS. The second images of (b), (c), (e) and (f) are obtained with the rotation of the first images by 45 degree. The white line in the image is indicating the director orientation. The particles in the images are different. The size of images is 40x40 µm2.
FIGURE 2. (a) Schematic diagram of the forces acting on a particle. The substrate surface is under the condition of strong planar anchoring energy along the director n. Forces from both the upper and bottom walls and buoyancy act on the particle. (b), (c) and (d) Side views of the particles in the cell. (c) and (d) are particle images of DC and QC respectively.
FIGURE 3. Calculated δc and measured δm position shifts of the particles. (a) CPMS with DC. (b) CPMS with QC. (c) GRYPMS with DC. (d) GRYPMS with QC. The small inset of each figure shows an optical microscope image of the corresponding defect configuration. The indices c and m are position shift calculated by R and h and measured by an optical microscope respectively.
FIGURE 4. (a) and (c) Movement of two DCs and two QCs in the cell as the function of time. (b) and (d) Time vs. distance and distance vs. speed graphs of dipole-dipole interactions and quadrupole-quadrupole interactions.
FIGURE 5. Schematic diagram of the forces acting on a DC. The dotted circles are the effective radius of a particle.
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polarizing optical images in nematic phase. (b) is DC texture of CPMS. (c) is QC texture of CPMS. (e) is DC
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Figure 2
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Figure 1
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Figure 4
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Figure 3
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Figure 5
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The novelty of the work The interactions of weak homeotropic anchoring particles were observed experimentally and analyzed with introducing the image charges and effective radius.
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Colour graphic
