* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download vapor liquid - UPB Sci Bull
Survey
Document related concepts
Photon scanning microscopy wikipedia , lookup
Rutherford backscattering spectrometry wikipedia , lookup
Harold Hopkins (physicist) wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
3D optical data storage wikipedia , lookup
Ultraviolet–visible spectroscopy wikipedia , lookup
Retroreflector wikipedia , lookup
Liquid crystal wikipedia , lookup
Photonic laser thruster wikipedia , lookup
Nonlinear optics wikipedia , lookup
Optical tweezers wikipedia , lookup
Transcript
U.P.B. Sci. Bull., Series A, Vol. 71, Iss. 2, 2009 ISSN 1223-7027 SURFACE INFLUENCE ON THE OPTICAL FREEDERICKSZ TRANSITION IN NEMATIC LIQUID CRYSTALS CELLS Eleonora - Rodica BENA1 , Cristina CÎRTOAJE2 Câmpurile critice pentru tranziţiile Freedericksz electrice, optice şi magnetice sunt puternic influenţate de condiţiile de ancorare ale moleculelor de cristal lichid la pereţii celulei. Utilizând o metodă de investigare bazată pe ecuaţiile Euler-Lagrange şi considerând că energia de ancorare la suprafaţă a moleculelor la interfaţa cristal lichid – perete este descrisă de formula Rapini-Papoular, am găsit o corelaţie între intensitatea fascicolului laser care produce destabilizarea alinierii homeotrope, energia de ancorare, grosimea celulei şi parametri de material. Am găsit de asemenea intensitatea de saturaţie a fascicolului laser care produce alinierea planară pe întreaga celulă, funcţie de proprietăţile cristalului lichid şi parametrii de suprafaţă şi grosimea celulei. The critical fields for electrical, magnetical and optical Freedericksz transitions are strongly influenced by the anchoring conditions of the molecules of the liquid crystal to the solid walls of the cell. Using an analytical method based on the Euler-Lagrange equations and considering that the surface anchoring energy of the liquid crystal-wall interface is described by Rapini-Papoular formula, we found a correlation between the value of the critical intensity of the laser beam producing the destabilization of the homeotropic alignment, the anchoring strength, the cell thickness and the material parameters. We also found the saturation intensity of the laser beam producing the planar alignment all over the cell, in terms of nematic liquid crystal properties, surface and cell parameters. 1. Introduction In confined structures of liquid crystals, the magnitude of the free energy terms associated with elasticity, surface anchoring and coupling to the applied external fields are frequently comparable. The critical fields for electrical, magnetical [1, 2, 3, 4] and optical [5] Freedericksz transitions are strongly influenced by the anchoring conditions of the molecules of the liquid crystals to the solid walls of the cell [6, 7]. It may be also changed in guest–host mixtures, such as those containing azo-dyes, when illuminated with U.V. light [8, 9]. 1 Prof., Physics Department, University POLITEHNICA Of Bucharest, Romania Assist., Physics Department, University POLITEHNICA Of Bucharest, Romania, e-mail: [email protected] 2 86 Eleonora - Rodica Bena, Cristina Cîrtoaje Rapini and Papoular have proposed a simple phenomenological expression for the anchoring free energy per unit area [6] 1 f s A sin 2 0 2 (1) where 0 is the angle between the easy direction e and the director n of the nematic liquid crystal at the nematic-wall interface. A is the so-called anchoring strength. It can be strictly proved that when the anchoring energy takes the form of Eq. [1], only a second order transition will occur [7]. By considering a new expression for the anchoring energy: fs 1 A sin 2 0 1 sin 2 0 2 (2) Yang et. al [10], [11], have shown that in the case of 0 the magnetic and respectively the optical-field induced Freedericksz transition in NLC cells are first order. Using Eq. 2 for the anchoring energy Bena and Petrescu [12] have studied the surface effects on magnetic Freedericksz transition in feronematics. In this paper we study the influence of the surface anchoring and the cell thickness on the optical-field induced Freedericksz transition in a homeotropic cell. 2. Theory 2.1 Basic Equation We consider an infinite flat layer of NLC between two identical substrates. The NLC molecules are initially homeotropically aligned to the transparent walls, considered parallel to the xOy plane and located at z d and z d . In the presence of a monochromatic laser beam, the NLC director can be reoriented: for p - polarization, the molecular director moves in the incidence plane xOz. We assume the laser beam having the vector k parallel to Oz and the electric field of the electromagnetic wave E parallel to Ox (Fig.1). Surface influence on the optical Freedericksz transition in nematic liquid crystal cells 87 Fig. 1 Orientation of the molecular director with respect to the laser field B and E are the magnetic and electric fields of the laser wave The free energy of the liquid crystal is: F S fv , z dz SA sin 2 0 d (3) d where S is the cell plates surfaces and fv , z is the free energy density in the bulk of the system in the presence of the laser field [4]. 1/ 2 1 I II (4) f v , z K1 sin 2 K3 cos 2 z2 1/ 2 2 a cos 2 Here K1 and K3 are the bend and splay elastic constant respectively, z d z , dz I is the mean volumic density of the electromagnetic energy of the light (connected with the intensity J of the light by relation J cI ), II and are the dielectric constants parallel and perpendicular to n respectively; a II is the dielectric anisotropy. In order to find the critical intensity of the laser field producing a damage of the homeotropic arrangement (i.e. the optical Freedericksz transition) we have to solve the Euler-Lagrange equation: d f v f v (5) 0 dz z with the boundary condition: d K1 sin 2 K3 cos2 (6) A sin 0 cos 0 dz z d In the middle of the cell ( z 0 ), 0 m , z 0 0 ; we assume the distortions to be symmetric: z z . As f v does not depend explicitly on z , a prime integral of Eq. (5) has the form: 88 Eleonora - Rodica Bena, Cristina Cîrtoaje fv z f v C1 z (7) Using Eq. (4), Eq. (7) leads to: I II 1 d K1 sin 2 K3 cos 2 C1 1/2 2 dz a cos 2 1/2 2 In z 0 , d dz (8) 0 and 0 m , so z 0 I II 1/2 C1 (9) 2 a cos m 1/2 Eq. (8) can be written: d dz 2 I II K1 sin 2 K3 cos 2 1/ 2 1 2 a cos m 1/ 2 (10) 1/ 2 a cos 2 1 a 1 , expanding into series the expression of the last bracket in Eq. (10), we obtain: Assuming I a II dz K3 1 sin 2 sin 3 a 3 a 2 2 1 2 4 sin sin m where: d 2 m sin (11) 2 K1 K3 K3 Integrating in Eq. (11) we have: I a II d K 3 1 sin 2 3 3 2 2 2 2 0 1 2 a 4 a sin sin m sin m sin Introducing the change: sin 0 sin sin ; sin 0 sin m sin m we obtain: d (12) (13) Surface influence on the optical Freedericksz transition in nematic liquid crystal cells d sin m 1 sin 2 1 sin 2 m sin 2 89 d and Eq. (12) becomes: I a II K3 1 sin 2 sin 2 2 d o d 3 a 3 a 1 sin 2 m sin 2 2 2 1 sin sin 1 m 2 4 (14) 2.2. The threshold intensity for Freedericksz transition In order to find the critical value of I that induces a slow transition from to 0 m 0 in the middle of the cell we have to consider m 0 in Eq. (14). We obtain: I c a II 2 d 0 K3 d 3 1 a 2 and, by integrating we get: I c a II K3 3 a 1 d 0 2 2 (15) From Eq. (6) and Eq. (10) and using the above mentioned expansion into series we get: I a II 3 a 3 a sin 2 0 sin 2 m K1 sin 2 0 K 3 cos 2 0 1 2 4 A sin 0 cos 0 sin 2 m sin 2 0 With the change (13), Eq. (16) becomes (16) 90 Eleonora - Rodica Bena, Cristina Cîrtoaje I a II 3 a 3 a sin 2 m sin 2 0 1 K 3 1 sin 2 0 sin 2 m 1 2 4 A sin 0 1 sin 2 m sin 2 0 cos 0 (17) Fig. 2. The threshold light intensity J c (W/cm 2 as a function of the anchoring strength A for optical Freedericksz transition. At the limit of the transition ( m 0 ) Eq. (17) becomes: I c a II 3 a 1 tan 0 2 Eliminating 0 from Eq. (15) and (18) we obtain: K3 E I c cot Ed I c A K3 A (18) (19) where E I c a II K3 3 a 1 2 (20) Eq. (19) gives a correlation between the value of critical intensity of the laser beam producing the damage of the homeotropic alignment, the anchoring strength A , the cell thickness 2d and the material parameters ( K3 , a , II , ). Surface influence on the optical Freedericksz transition in nematic liquid crystal cells 91 In the rigid anchoring limit A and considering a we obtain K3 the formula for optical Freedericksz transition in a 4 d a II homeotropic cell [13], [14]. I C 2 2 2.3 The saturation intensity If the laser beam intensity is high, it can produce the transition from homeotropic to planar alignment all over the cell, i.e. the saturation of the Freedericksz transition. In order to find the saturation intensity I s producing the distortion m in the middle of the cell, we reconsider Eqs. (12) and (16) with the 2 change of variables: cos m cos m (21) cos ; cos 0 cos cos 0 Eq. (12) becomes: I a II 0 d 0 cos 2 m K 3 1 1 cos 2 3 a 3 a cos 2 m 2 1 cos 2 1 2 cos 2 4 d cos cos 2 m 2 (22) When m 2 we have: I s a II d 0 0 K1 d cos (23) By integrating: I s a II From Eq. (16) we get: d K1 1 sin 0 ln 2 1 sin 0 (24) 92 Eleonora - Rodica Bena, Cristina Cîrtoaje I s a II K1 A sin 0 (25) Eliminating 0 from Eqs. (24) and (25) we obtain the relationship: I I (26) A coth s a II K1 s a II K1 which enables us to calculate the saturation intensity of the laser beam as a function of the material parameters and the anchoring strength A . In the rigid anchoring limit A there does not exist a laser intensity that produces the saturation transition. For material parameters we can use the following values: K 3 7.4 1012 N, K1 5.3 1012 N, II 2.89 , 2.25 , ( a 0.64 ). The cell thickness is 2d 2 104 m and the anchoring strength A is in the range 107 103 J/m 2 , [15], [16]. The threshold and the saturation intensities of the light obtained from Eqs. (19) and (26) respectively, as a function of the anchoring strength A are shown in Fig. 2 and Fig. 3 respectively. Fig. 3 The saturation light intensity for optical Freedericksz transition as a function of the anchoring strength. Surface influence on the optical Freedericksz transition in nematic liquid crystal cells 93 3. Discussion As can be seen in Fig .2, the critical intensity J C for optical Freedericksz transition is slightly influenced by the anchoring strength: when A increases hundred times J C increases only three times; for values of A greater than 5 106 J/m 2 the threshold intensity is practically constant and equal to the value obtained in the case of the rigid anchoring. Regarding the saturation intensity, it increases rapidly with A ; for A of the order of 3 107 J/m 2 the laser intensity exceeds the acceptable value for a liquid crystal cell experiment. 4. Conclusion We have studied analytically the laser -field induced Freedericksz transition in an homeotropic aligned NLC, taking into account the surface anchoring effects. We have calculated the threshold intensity of the laser beam producing the transition and the saturation intensity as a functions of the anchoring strength. For obtaining the Fredeericksz transition using convenient laser intensities it is possible to take a large interval for the anchoring strength ( 107 104 J/m 2 ) while for obtaining the saturation, the anchoring strength must be small ( 107 J/m 2 ). Our results can be useful in designing of practical devices used in optoelectronics. Our results can be useful in designing of practical devices such as light shutters or laser writing devices. REFERENCES [1] E. Petrescu, Cornelia Moţoc, Doina Mănăilă, Mod. Phys. Lett. B, 14 (4) (2000) 139. [2] Cornelia Moţoc, E. Petrescu, Mod. Phys. Lett. B, 12 (13), (1998) 529. [3] Cornelia Moţoc, E. Petrescu, V. P. Păun, U.P.B. Sci. Bull. A, 61 (3-4) (1999) 135. [4] Cornelia Moţoc, E. Petrescu, U.P.B., Sci. Bull., A 59 (1-4), 1997, 73. [5] E. Petrescu, Cornelia Moţoc, Camelia Petrescu, Mol. Cryst. Liq. Cryst. 415 (2004) 415. [6] A. Rapini and M. Papoular, J. Phys. (Paris) Colloq. 30 (1969) C4-54. [7] H. J. Dewling, Mol. Cryst. Liq. Cryst., 19 (1972) 123. [8] Gabriela Iacobescu, A.L. Păun, Cristina Cîrtoaje , J. Magn. Magn. Mater. 320 (2008) 1280. [9] Cornelia Moţoc, Gabriela Iacobescu, J. Magn. Magn. Mater. 306 (2006) 103. [10] G. C. Yang, J. R. Shi and Y. Liang, Liq. Cryst. 27 (2000) 875. [11] J. R. Shi and H. Yue, Phys. Rev. E 62 (2000) 689. [12] Eleonora Rodica Bena and E. Petrescu, J. Magn. Magn. Mater. 263 (2003) 353. [13] P. Oswald, P. Pieranski, Nematic and cholesteric liquid crystals, (Taylor & Francis, Taylor & Francis Group, Boca Raton, London, New York, Singapore, 2005). 94 Eleonora - Rodica Bena, Cristina Cîrtoaje [14] Cornelia Moţoc, Gabriela Iacobescu, Liquid Crystals Physical Properties and Applications, (Universitaria Press, Craiova , 2004 (in Romanian)). [15] L. Ong, Phys. Rev. A 28, 2393 (1983). [16] Iuliana Cuculescu, Rodica Bena, G. Smeianu, Liq. Cryst. 4(4), (1987) 457.