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Molecular arrangements in liquid crystalline mesophases. In a nematic mesophase, molecular orientations are correlated, while molecular positions are not. The average orientation is termed a director, n. In a cholesteric mesophase the average molecular orientation twists through the medium with a certain periodicity, while positions of molecules are not correlated. In a smectic A mesophase molecules lie in planes. Molecular axes are perpendicular to these planes but otherwise are not ordered within the planes. Smectic B has a hexagonal packing of molecules in the planes, while in smectic C the director is tilted in the planes. Columnar mesophases are often formed by disc-shaped molecules. The most common arrangements of columns in two-dimensional lattices are hexagonal, rectangular, and herringbone. In the herringbone mesophase molecules are tilted with respect to the columnar axis.
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This pedagogical overview of liquid crystals is based on lectures for postgraduate students given at the International Max Planck Research School "Modeling of Soft Matter". I am delighted to dedicate it to my scientific advisor, Prof. Yuriy Reznikov, thus acknowledging his valuable contribution to my life.
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... on the arrangement of the molecules in a mesophase, or its symmetry, liquid crystals are subdivided into nematics, cholesterics, smectics, and columnar mesophases. Molecular arrange- ments of these mesophases are depicted in Fig. ...
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... of their ends, even if they differ. Hence, the sign of the director has no physical significance, and the nematic behaves optically as a uniaxial material with a center of symmetry. We will introduce a mathematically rigorous definition of the director in Section 4. The director and the molecular arrangement in a nematic mesophase are sketched in Fig. 1, where the anisotropic shape of molecules is depicted by ...
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... The director distribution is precisely what would be obtained by twisting a nematic aligned along the y axis about the x axis. In any plane perpendicular to the twist axis the long axes of the molecules align along a single preferred direc- tion in this plane, but in a series of parallel planes this direction rotates uniformly, as illustrated in Fig. 1. The secondary structure of the cholesteric is characterized by the distance measured along the twist axis over which the director rotates through a full circle. This distance is called the pitch of the cholesteric, p. The periodicity length of the cholesteric is actually only a half of this distance, since n and −n are ...
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... in their positions in addition to the orientational ordering. The layers can slide freely over one another. Depending on the molecular order in layers, a number of different types of smectics have been observed. In a smectic A, molecules are aligned perpendicular to the layers, without long-range crystalline ordering within them, as shown in Fig. 1. In a smectic C, the preferred molecular axis is not perpendicu- lar to the layers, so that the phase has biaxial symmetry. In a smectic B, there is a hexagonal crystalline order within the ...
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... placed between glass substrates, smectics do not assume the simple arrangement shown in Fig. 1. To preserve their thick- ness, the layers become distorted and can slide over one another in order to accommodate the substrates. The smectic focal conic texture appears due to these distortions. Typical textures formed by smectics are shown in Fig. ...
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... are grouped according to the packing motive of the columns. In columnar nematics, for example, molecules do not form columnar assemblies but only float with their short axes parallel to each other. In other columnar liquid crystals columns are arranged in two-dimensional lattices: hexagonal, tetragonal, rectangular, and herringbone, as shown in Fig. ...
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... and is very similar to the distribution function of the Maier-Saupe theory. With the help of this ansatz, the dependence of the scalar order parameter S on the parameter a can be written as The numerical integration of the double integral over h 1 and h 2 provides the dependence of the translational entropy on the order parameter and is shown in Fig. 10, together with the rotational entropy. As expected, the two contributions compete: in an ordered system with S ≈ 1 the rotational entropy is at its minimum, while the translational entropy is at its maximum, since the excluded volume is the smallest for c 12 = ...
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... inset of Fig. 10 illustrates the dependence of the free energy on the order parameter for three values of the dimensionless density, qLD 2 . If q is small, F has only one minimum at a = 0, which cor- responds to the isotropic state. Above q I , another minimum appears which corresponds to a nematic state. If q exceeds q N , the minimum at a = 0 ...
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... experimental setups, a liquid crystal sample is often placed between crossed polarizers. We adopt this geometry and assume that the first polarizer forms an angle a with respect to the director, as depicted in Fig. 11. The linearly polarized light after this polarizer, becomes elliptically polarized after passing through the ...
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... illustrate the director response to the external magnetic or electric fields, let us consider a nematic liquid crystal oriented by two glass substrates, as shown in Fig. 12. The interaction between the nematic and the substrates is such that the director is aligned along the substrate normals. Experimental observations tell us that if a magnetic field is applied perpendicular to the director and its magnitude exceeds a certain critical value, the optical properties of the system change abruptly. The ...
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... director profiles as well as exact and approximate dependen- cies of h m on H are shown in Fig. ...
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... transition is easy to observe optically, since the average refractive index of the material changes when the magnetic field is applied. For the light beam polarized along the x axis in the geometry shown in Fig. 12, the local refractive index reads n(z) = n e n o n 2 e sin 2 h + n o cos 2 h 1/2 , and the average difference in the optical lengths of the ordinary and extraordinary waves is Using Eq. (15), this can be rewritten ...
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... of the experimental methods measure surface director deviations in an external field and involve rather complicated opti- cal setups. One of the simplest measurements of weak azimuthal anchoring strengths can be performed in a wedge cell with a twisted distribution of the director [60], as shown in Fig. 14. The easy axis on the reference substrate, where strong anchoring is assumed, makes an angle a with respect to the easy axis on the test substrate. The director orientation on the test substrate, 0 t , can be found from the condition sin 0 t = n 2 sin 2(a − 0 t ), where n = Wd/k 22 , and d is the cell thickness. In a wedge cell d ...
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... respect to the easy axis on the test substrate. The director orientation on the test substrate, 0 t , can be found from the condition sin 0 t = n 2 sin 2(a − 0 t ), where n = Wd/k 22 , and d is the cell thickness. In a wedge cell d varies in the range ∼0-50 lm. For a = p/4 the explicit solution to this equation is sin 0 t = 1 2 2 + n −2 − n −1 . Fig. 14. Geometry of the wedge cell. The angle between the easy axis on the reference (er) and the test (et) substrates is a. The director n deviates from the test substrate easy axis by an angle 0 t ...
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... is shown as a function of the cell thickness d in Fig. 15 for differ- ent values of anchoring strengths. This dependence can, of course, be measured in a single wedge cell. Though this method is only suitable for anchoring strengths of up to 10 −2 erg/cm 2 , its range and accuracy can be improved by using a magnetic field to control deviations of the director from the orientation of the easy ...
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... of disclinations for several m are shown in Fig. 16. The elastic energy per unit length associated with a disclination is pKm 2 ln(R/r 0 ), where R is the size of the sample and r 0 is a lower cutoff radius, or the size of the disclination core [2]. Since the elas- tic energy is proportional to m 2 , the formation of disclinations with large Frank indices m is energetically ...
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... particles with such defects interact as a pair of dipoles. The other defect type is a −1/2 strength disclination ring that encircles the particle, or a Saturn-ring defect. This defect accompanies small colloids, has quadrupolar sym- metry, and results in quadrupole-quadrupole interactions between colloids. Both types of defects are sketched in Fig. ...
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... an illustrative example, a simulation snapshot of two colloids immersed in a Gay-Berne liquid crystal is shown in Fig. 18. The mean Table 1 Parameterizations of the Gay-Berne potential. 3 5 2 1 [143,144] 3 5 1 3 [145] 3 5 1 2 [146] 3 - 0 0 Soft repulsive potential 4.4 20 1 1 [147] interaction force between these colloids shows that the depletion forces dominate for small colloidal particles. The tangential compo- nent of the force can be used to resolve ...
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... single pixel of a twisted nematic display consists of two polar- izing filters, two transparent electrodes, and a reflecting mirror, as depicted in Fig. 19. The liquid crystal itself is sandwiched between the transparent electrodes and is aligned by the substrates in such a way that there is a p/2 twist of the director in the cell. Before entering the liquid crystal, the incident light is polarized by the first polarizer. In the liquid crystal, light polarization follows the director ...
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... Depending on their molecular symmetry, they are categorized as nematic, smectic, and cholesteric phases, where the nematic phase possesses a long-range orientational order while the smectic has some positional order along the orientational order. [2][3][4][5][6] LC-based display devices deal with certain issues such as image sticking, slow responses, high threshold voltages, and reduced voltage holding ratio due to the presence of mobile ions that degrade their performance. When an electric field is applied to these systems, the ions move towards the electrode surface, are adsorbed there, and thus create their own field which reduces the effect of the applied electric field. ...
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... 50,53,54 At second order and for uniaxial particles, this expansion gives rise to the nematic tensor Q ij , which is a symmetric traceless tensor. The eigenvalue S of Q ij with the largest absolute value measures the degree of nematic order; the corresponding eigenvector is the nematic director ⃗ n. 55 If the two other eigenvalues are equal to −S/2, the system is in an uniaxial (nematic) phase. On the other hand, if the system has three distinct eigenvalues, it is in a biaxial phase. ...
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... For random order by thermal averages gets cancelled, Optically, these are uniaxial materials and the magnetic susceptibility is also of the uniaxial form. Considering Eq. (3), it can be stated that, where n, called the director, is a unit vector parallel to the optic axis and S is the scalar order parameter, δ is the Kronecker delta 20 . ...
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