Sıvı kristallerde smetrik a-smetrik C* faz geçişinin klasik ve genelleştirilmiş landau modelleri ile incelenmesi
Classical and extended landau models introduction
- Tez No: 66822
- Danışmanlar: PROF. DR. HAMİT YURTSEVEN
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Faz geçişleri, Landau modeli, Sıvı kristaller, Phase transitions, Landau model, Liquid crystals
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Fizik Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada SmA-SmC* ferroelektrik faz geçişi için, Klasik ve Genelleştirilmiş Landau modelleri incelendi. Her iki model içinde serbest enerji yoğunluğunun Landau açılımı, iki bileşenli düzen parametreleri \ = {l>^,%2) tilt vektörü ve P = (Px, Py) polarizasyon vektörü cinsinden yazılmıştır. Bu modellerin kullanılmasıyla P polarizasyonun, 9 tilt açısının, P/9 oranının, p helisel adımın sıcaklığa bağlı değişimleri incelenmiştir. Ayrıca magnetik ve elektrik alanın sıcaklığa bağlı değişimleri Klasik model için incelenmiştir. Yapılan bu çalışma sonucunda Genelleştirilmiş modelin deneysel çalışma ile uyum içinde olduğu, Klasik modelin ise deneysel çalışmayı doğrulamadığı ortaya çıkmaktadır. IV
Özet (Çeviri)
SUMMARY The order parameter of the smectic A (SmA) to smectic C (SmC) transition is a two component tilt vector \ = (^, Ç2) describing the magnitude and the direction of the tilt of the long molecular axis from the normal to smectic layers. For chiral systems the tilt precesses around the normal to smectic layers as one goes from one layer to another resulting in a helicoidal Smectic C* (SmC*) phase. The tilt of a molecule breaks the axial symmetry around its long axis inducing an inplane polarization P = (px,Py) perpendicular to the tilt. Thermodynamic properties of the system are usually described by the phenomenological Landau type free energy, including the bilinear coupling between the primary order parameter \ and the secondary order parameter P. This coupling is of a chiral character and does not exist in nonchiral systems where no ordering of molecules transverse to their long axes is induced by the tilt according to the model. A. CLASSICAL MODEL A phenomenological description of SmC* ferroelectric liquid crystals was first introduced by Pikin and Indenbom [1]. The predictions of Classical model do not agree with experimental data especially in a narrow temperature interval below the SmA-SmC* phase transition temperature Tc. Their Classical model is as described below: x2 /“ N2 (D Classical model consists of the Landau expansion of the free energy density in the two, two-component order parameters \=^,Z)2) tilt vector and P = (Px, Py) polarization vector. The coefficient a of the quadratic term in the expansion in the primary order parameter | is linearly temperature dependent a = a(T-T0) and it becomesnegative at low temperatures corresponding to the instability of the untilted SmA phase. The fourth order term for b>0 stabilizes the tilted phase. The Lifshitz term in Ş (the A -term) is characteristic for chiral molecules and it produces, in competition to the elastic energy (the K3-term), the helicoidal structure. The polarization part of the free energy density consists only of the quadratic term with a positive coefficient (e>0), as the dipole-dipole interactions are not expected to be large enough to influence the SmA-SmC* phase transition. For the same reason derivatives of P do not appear in the expansion. The polarization is therefore only locally induced by the tilt because of two bilinear coupling terms. The piezo-electric bilinear coupling (C-term) is of a chiral nature and it is a consequence of a mirror plane symmetry broken by the tilt. The flexo-electric bilinear coupling (ji-term) gives the polarization induced by the gradient of the tilt. This coefficient is not of chiral character and jj, * 0 for nonchiral systems, but in this case gradient of \ equals zero and the flexoelectric coupling has no effect. Therefore for nonchiral systems in the SmC phase there is no coupling between the tilt and the polarization and P = 0. Polarization exists only in the chiral SmC* phase as required by symmetry. Using the order parameter components 4, =9cosqz, Ç2 =0sinqz Px = -Psinqz, Py = Pcosqz one obtains by minimizing the free energy (1) with respect to the tilt 0, polarization P and the wave vector of the helix q, respectively, for T<TC P = -^%r (2) A + jisC and P = GsC (3) where s= st<3 ”and 0 = 0 + -^ K3-s(a2 K 3 The tilt angle can be obtained as 2 e = ± f«(Tc-T) (4) where the transition temperature Tc is given by TC=T0 + a,( 7^,2 A' ^ K3y VIWithin this model the pitch of the helix p does not depend on temperature in contrast to experiments. The spontaneous polarization is strictly proportional to the tilt and both show a classic square root temperature dependence. Eq. (3) can used to find P / 9 as given below: E = sC (5) P Also the ratio - does not depend on temperature in contrast to experiments. Using expression (1) and eliminating Px and Py by defining Px =sp^-sCÇ2 Py =6^+80^ one finds in the Classical model the free energy density g(z): g(z) = |(a-sC2)02+|be4-02(t,'A + ^K302(t.'2 (6) Here A = A + s^iC and K3 = K3 - sjj.2 In the presence of a magnetic field H applied in a direction perpendicular to the 1 helical axis, a term gH = -:rXaH2Ç2 has to be added to the free energy density (6). The magnetic field tends to align the molecules because of their magnetic anisotropy %a = %n - %x and deforms the helix. The free energy has to be in the constant amplitude approximation minimized with respect to <|)(z) leading to the Sine-Gordon equation dz2 f,.2\ %aH v K3 j ?sin 2$ (7) which admits non-linear phase soliton solutions for H * 0. For H>HC the modulated structure is not stable and the system makes a transition into the homogeneously tilted SmC phase. The SmC*, SmC and SmA phases coexist at the Lifshitz point which is given by 1 A2 A2 TL =T0+-eC2+4^- and HL = 2 ' a aK3 "VXaK3 <8> The magnetic field dependence of the SmA-SmC* transition line is obtained by using the Eqs. (7) and (8) as, VIITC-T = AT W_ H4 ' HL2+4H4J Here AT = -4r- aK3 In the presence of a electric field E, a applied perpendicularly to the helical axis, a term gE = -EPy has to be added to the free energy density (6). Minimizing the free energy with respect to <j)(z) leading to the Sine-Gordon equation d2(|> _EPsin(|) dz2 ~ K392 which gives as a solution a 2it- soliton lattice. The period of this lattice goes to infinity at a critical field ?=. = ** (9) PK3 This is different from the magnetic case where a tc- soliton lattice appears because of the quadratic coupling to the order parameter. According to (9) the temperature dependence of Ec should be the same as of the tilt 8 oc (Tc - T)n. The electric field dependence of the SmA-SmC* transition line is as given below: TC-T = AT ( 4E &\. + -c ^ Ec E2J B. EXTENDED MODEL All chiral terms in (1) are expected to be small. This can be concluded from the large value of the pitch as compared to the molecular dimensions and from the extremely small difference in the transition temperatures Tc-T0 between a chiral system and a corresponding racemic mixture. As the bilinear P - e coupling terms are chiral and therefore small, the biquadratic coupling terms could become important in SmC* phase. On the other hand, NMR measurements show no appreciable difference in the transverse ordering between chiral and non-chiral systems except perhaps very close to Tc. The observed quadrupolar ordering can be described by P2* 0 and it is induced by the biquadratic P - s coupling terms. The lowest nonchiral biquadratic coupling terms are added to the free energy expansion (1). VIIIga=-ln(P^2-Py^1)2+]tl(Px2+Py2)2 The last term has been added to stabilize the system. The higher order term can be added gb=-d(§2+^1 dz -S: d*i' dz gb is equivalent to replacing A by A + d82 and this should describe the monotonous increase of the pitch with temperature at low temperatures. The sixth order term in tilt has been added to account for the temperature dependence of the specific heat of the system. The total free energy g+ga+gb+gc can then be expressed with the components of the order parameter ^ = Bcosqz, l,2 = Gsinqz Px =-Psinqz, Py =Pcosqz as a function of 0, P and q as +^(Px2+P*V(Px^ (1°) dz) \dz. Minimization with respect to q gives for the wave vector of the helix q = K, de2+-Bp-+A 9 (11) 2n The temperature appears in q and therefore in the pitch p = - in two ways: q 1) d-term gives a contribution at large 9 (low T) and it describes the low temperature increasing part of p(T) for d>0, 2) the last term is proportional to the flexoelectric coupling coefficient n and it is temperature dependent only if p is not proportional to 9. IXBy eliminating q from (10), one obtains the free energy as a function of Band P. Then minimizing the free energy g(z) with respect to P leads to a cubic equation for P TlP3+P 1 Vs -2-R 2^ ( A^ uB2d^ -9 C + -^ + 3/ V K, K 0 (12) 3 / u 1 1 n Here - = --.*- s s K and C = C + A|i k7 One solution to Eq. (12) is given by P = 2ti ~ uö2d^ c+- - K3 J 18 3 1 M 2^ + - -~+a© 3ti J e f~ u92d^ Vz C+.fr; - + 2ti K 3 / 18 (13) In Eq. (13) z = 4K2. -12K2Q92e + 12K^29462 -4K2Q306e3 + 27K2r|C202e; + 54K3TiC04s3nd + - 3 _2 27V3~e66V2^d2 K, Eliminating q and P from Eq. (10) and minimizing this free energy g(z) with respect to 9, we obtain the equation for 0 as K,U03 Sd^c ePc+be3 +9 a-Q 0 2n fc neRch Vz' c+ ^ + 18 J+.1f_]+nepT+2f-!l+ner; 9rfV s J 3rf s 9 2q C+ jaePdl Vz K 3 J 18 0 2n, K3J 18 =0 Here a = a A^ K, ~,_ 4Ad _ 3d2 b = b- - -, c = c- K, K, (14)The temperature dependence of 0 can be obtained numerically from Eq. (14). Therefore the temperature dependences of P and p can be calculated from Eqs. (11) and (12), respectively. XI
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