Sıvı kristallerde smektik a-smektik C faz geçişinin P2O2 çiftlenimli landau modeli ile incelenmesi

Abstract

ÖZET Sıvı kristal karışımlarda Smektik A - Smektik C* faz geçişi ortalama alan teorisine göre incelenebilir. Ortalama alan teorisinde serbest enerji düzen parametreleri yönelim açısı 9 ve polarizasyon P nin kuvvetleri cinsinden yazılabilir. Bu çalışmada serbest enerji açılımında polarizasyonla yönelim açısı arasındaki P`6` çiftlenimi ile yönelim açısı ve polarizasyon sıcaklığın fonksiyonu olarak Smektik A - Smektik C* faz geçişi için hesaplanmıştır. Yönelim açısının sıcaklığa bağlılığı deneysel olarak gözlenen değerlere fit edilerek serbest enerji parametre değerleri hesaplanmıştır. Bu parametre değerleri kullanılarak polarizasyonun sıcaklığa bağlılığı hesaplanmış ve deneysel değerler ile karşılaştırılmıştır. P2 02 çiftlenimine göre bu modelden hesaplanan polarizasyonun sıcaklığa bağlılığı deneysel sonuçlarla uyum içindedir. IV

PHASE TRANSITION BETWEEN SMECTIC A AND SMECTIC C* PHASES EN FERROELECTRIC LIQUID CRYSTALS SUMMARY In this work, we have studided phase transition between Smectic A and Smectic C* phases in ferroelectric liquid crystals. Until 1986 the experiments have shown that Smectic A - Smectic C* phase transition is considered as a second order phase transition [1-7]. Since then, experiments on different ferroelectric liquid crystals have shown a first order phase transition between Smectic A and Smectic C* phases [8-13]. Some ferroelectric liquid crystals also show a tricritical şoint [14-18]. Smectic A - Smectic C* phase transition can be studied using the Landau mean field theory. In this mean field theory the free energy can be expanded in powers of the order parameters, tilt angle 0 and polarization P. In the literature P0 coupling term in the free energy expansion has been studied [12], In our work, we took the coupling term as P^ö`. In both cases the coupling terms have been taken in a way that the free enery is reduced. In the Smectic C* phase we can take P * 0, 9 * 0 and in the Smectic A phase we have P = 0, 0 = 0. In our work we expand the free energy in terms of order parameters P and 0 as : g=-aO- +-b04 +-c0b+-P--DP-0-+-eP* (1) 2 4 6 2;r0*0 4 where Xo is the susceptibility and s0 is vacuum permittivity. Here we have added the ~eP4 term in order to satisfy thermodynamic stability. In equation (1) we have a = a (T - T0) and the coefficient b is taken as a negative quantity in order to satisfy that the transition is of first order. Here we take the coefficients c, D, e as positive parameters. From the rninimization of the free energy given in equation (1) with respect to 0, we obtain that aO + bO3 +c0* -2DP2Û = 0 (2) From the minimization of the free energy given in equation (1) with respect to P, we getXoso P-2DP02 +eP = 0 (3) From equation (3) the polarization can be obtained as P = (ids2 1 e ez0s0J (4) Inserting equation (4) into equation (2) gives c04+b*02 +d*=0 (5) Here we have b* = b- 4D2 (6) d* = a(T-Tn)- 1D_ eZo£o (7) Inserting the polarization given by equation (4) into the free energy given by equation (1) gives g=^a'02 + ~b'04 +/c'66 (8) Here we have a =a + 2D (9) VIb' = b + - AD2 (10) and c' = c. From the first order condition we can write a'c' 3 b'2 ~~ 16 (11) Since the temperature under study is the experimental phase transition temperature T^. in equation (9), we have a = a (Tc- T0). Using equations above, we get, ba 2D T. = T0+- -16 ac eaZo£o (12) Using T0 given in equation ( 12), we can rewrite equation (7) as 3 V2 (13) Inserting equation (13) into equation (5) gives aAT = -c04 + r4D2 >-b e2- V e J 16c 4D2 V- b - K e J (14) Here AT = T - Tc and T0 is the experimentally observed critical temperature. From the fitting of 9 versus AT to the experimental values given by Bahr, Heppke and Sabaschus [12] we get the coefficients a, b, c, D, e given in Table 1. In Figure 1 we show the fitting of 9 versus AT. The dependence on temperature of the polarization can be obtained from equation (4) and ( 14) by taking the susceptibility Xo = 4. Our theoretical values and those experimental given in reference [12] for the polarization are given in Figure 2. vuTable 1. The values of the parameters given in the free energy. 2.65 x 107MKS -1.67 x 10s MKS 2.09 x l(f MKS D 7.45 x 10` MKS 17 1.20 x 10` MKS Figure 1. Temperature dependence of the tilt angle vinFigure 2. - Temperature dependence of the polarization IX