Sıvı kristal ikili karışımlarında Halperin-Lubensky-Ma ölçeklenme fonksiyonunun çift kırıcılık ölçümleri ile test edilmesi
Testing Halperin-Lubensky-Ma scaling function in liquid crystal binary mixtures via birefringence measurements
Yazar: ŞENAY ÜSTÜNEL
Danışman: PROF. DR. SEVTAP YILDIZ ÖZBEK
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Fizik Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2017
Konu: Fizik ve Fizik Mühendisliği = Physics and Physics Engineering

Özet:
Sıvı kristal hali maddenin ara hallerinden biridir. Sıvılar izotropiktir yani her yönde aynı özellikleri gösterir. Kristal katılarsa anizotropiktir, birçok fiziksel özelliği yöne göre farklıdır. Sıvı kristal fazında ise birçok fiziksel görünüm olarak sıvıya benzeseler de, molekülleri aniztropik özellikler gösterecek şekilde düzenlidir. Sıvı kristaller iki çeşittir, birincisi liyotropik sıvı kristaller, diğeri ise termotropik sıvı kristallerdir. Liyotropik sıvı kristaller ara fazlarını çözücü konsantrasyonuna bağlı olarak gösterirler. Termotropik sıvı kristaller ise adından da anlaşılacağı üzere sıcaklığa bağlı olarak ara fazlar gösterirler. Bilimsel çalışmalarda termotropik sıvı kristaller en çok kullanılanlardır. Termotropik sıvı kristal moleküllerin konumsal yönelimsel düzenine göre farklı gruplara ayrılırlar. Temel olarak üç ana ara fazı bulunmaktadır; nematik, smektik ve kolesterik. Nematik sıvı kristaller uzun erimli yönelimsel düzene sahiptirler ve tercihli bir n ̂ (direktör) vektörü yönünde yönelirler. Smektik sıvı kristaller ise hem konumsal hem yönelimsel düzene sahiptirler ve katmanlı yapılarıyla karakterize edilirler. Tabaklar arası ve tabakanın içindeki korelasyona bağlı olarak smektiklerin farklı türleri vardır, keşfedildikleriyle sırayla alfabetik olarak (Smektik A, B, C…) kodlanmışlardır. Smektik A (SmA) fazındaki moleküller katmanlarda birbirine paralel olarak yönelmişlerdir, bu yönelim direktör yönündedir. Kolesterik sıvı kristaller kiral nematik sıvı kristallerdir, yani nematiklerden tek farklı moleküller sarmal şeklinde katmanlar halinde yönelmiştir. 
Faz geçişi maddenin dışarıdan bir etkiyle (sıcaklık ve basınç) bulunduğu fazdan başka bir faza geçmesidir. Faz geçişinde iki fazın birbirinden ayırt edilemediği noktaya "kritik nokta" buradaki sıcaklık ise "kritik sıcaklık" olarak adlandırılır ve T_C ile gösterilir. Eğer bir faz geçişinde gizli ısı ortaya çıkıyor ve entropinin süreksiz olarak değişiyorsa, bu birinci dereceden faz geçişi, entropi sürekli ve gizli ısı gerekmiyorsa ikinci dereceden faz geçişi olarak adlandırılır. Sıvı kristaller birçok ara faza sahiptir. Bu nedenle faz geçişleri konusunda birçok model ve teorilerin geçerliliği sıvı kristaller kullanılarak gösterilebilir. Bu çalışmada sıvı kristallerin bir anizotropik özelliği olan çift kırıcılık kullanılarak Nematik – Smektik A (N-SmA) faz geçişi civarında Halperin-Lubensky-Ma (HLM) teorisiyle uyumu incelenmiştir. Çift kırıcılık anizotropik malzemeden giren ışığın iki farklı ışığın kutuplanmasıdır, ∆n ile ifade edilir. Bu çalışmada çift kırıcılık ölçümleri yüksek hassasiyet özellikli döner analizör metotu yapılmıştır. 
Çift kırıcılık verileri kullanılarak nematik düzen parametresi daha önce Özbek ve ekibi tarafından gösterilen S(T)=Δn(T)/〖Δn〗_0  bağıntısıyla bulunmştur. Anisimov ve ekibi tarafından N – SmA geçişindeki düzen parametresi ψ ve direktör dalgalanmasının serbest enerjiye eklenmesiyle bu ifadeye kübik bir ifade eklendiğini göstermiştir. Bu kübik terim geçişin birinci derece olmasını gerektirir. HLM teorisinde de N – Sm A geçişi birinci derece olması gerekir. Daha sonra evrensel ölçeklenme fonksiyonunu konsantrasyon x ve S ̅ indirgenmiş entropi değişimi bağlı olarak S ̅/S ̅^* -(S ̅/S ̅^* )^(-1/2)=a ̂/S ̅^*  (x-x^* )≡y-y^* ifadesiyle verilmiştir. Çetinkaya ve ekibi bu evrensel ölçeklenme fonksiyonundaki entropi değişimiyle N- SmA geçişinde nematik düzen parametresi S ilişkilendirmiştir. 
Bu çalışmada 8CB – 10CB ikili karışımlarında çift kırıcılık ölçümlerinden nematik düzen parametresi S, türetilmiştir. Bu S verileri T_NA N –SmA geçiş sıcaklığı civarında olası entropi süreksizliklerine ulaşmak için kullanılmıştır. Elde edilen mol kesri x'e bağlı entropi süreksizlikleri evrensel ölçeklenme fonksiyonuna iyi bir şekilde fit edilmiştir. Bu da HLM teorisindeki smektik A ve yönelimsel düzenin çiftlenimden dolayı kübik terim eklenen ortalama alan ifadesiyle tutarlıdır. Ayrıca daha önce yapılan kalorimetrik ölçümlerle bulunan sonuçlarla oldukça uyumludur. Daha sonra T_NA civarındaki S(T)'nin sıcaklık türevinin aynı kuvvet yasası ıraksaklığı (özgül ısı kapasitesi) olduğunu kullanılarak etkin kritik üstel değerleri bulunmuştur.

Abstract:
States of matter can be grouped into liquid, solid and gas in general. However, liquid crystal phase have been found as a mesophase. Liquids are isotropic which means they exhibit same property in every directions, but crystalline solids are anisotropic, that is they show different properties in different directions. Liquid crystals exhibit the physical features of an ordinary liquid, but while their molecules are ordered leading to anisotropy. Therefore, their optical, electrical properties vary with direction of these ordered units. Liquid crystal phase is first discovered by a plant physiologist in 1888. 
Liquid crystals divided into lyotropic and termotropic liquid crystals. Lyotropics exhibit their mesophases with the change of solvent concentration. Thermotropic liquid crystals show their intermediate phases with temperature change. Thermotropic liquid crystals have wide-range study and usage area. They are divided in subclasses with respect to locational and orientational order of their molecules. Basically, they exhibit three distinct mesophases; nematic, smectic and cholesteric. In the nematic phase molecules have long-range orientational order and they are oriented with respect to preferred direction the so-called director and designated by unit vector n ̂. Their center of the mass are distributed randomly. Molecules tend to align parallel. For adjusting the director in desired direction can be made by chemical or/and physical treatments. Molecules can be aligned by planar which is parallel to substrate or homeotropic which is perpendicular to surface. Molecules in the smectic phase are arranged in layers, correlations in and between the layers define types of smectic phase. According to order of the discovery, they are coded alphabetical. Smectic A phase is the most used and common smectic phase. Molecules form layers in the direction of n ̂. Layers in the smectic A phase can slide over one another, liquid crystals in this phase behaves like a two dimensional liquid. There is still orientational and molecules are packed tighter. Cholesteric liquid crystals are called as chiral nematics. Their all physical properties are same with nematic liquid crystals except their molecules are spiral ordered to make layers, this property comes from chiral molecules added to nematic liquid crystals and cholesteric liquid crystals are synthesized. Cholesteric liquid crystals have more symmetrical order than nematic liquid crystals and they form a helical structure in the direction of director. The most important property in cholesteric liquid crystals is spiral pitch which is the distance the director took with a full turn. This spiral pitch can be changed via external sources like temperature, magnetic field, etc. Since their spiral structure they can reflect light in wavelength equal to spiral pitch. They are widely used in different types of sensor. 
Phase transition is said to be the change of matter's present phase under the change of temperature and pressure. The point at which one cannot distinguish two phases with each other is called the critical point and temperature at that point is named critical temperature T_C. In a phase transition if there is latent heat and there is a discontinuity in entropy change the phase transition is of first order. If there is no latent heat and entropy is continuous this phase transition is of second order. Liquid crystals are known to be good candidates in order to determine and test of different phase transition models and theories in statistical physics, because of their variety of phases and their phase transition temperatures can be reached at laboratories easily. Intrinsic properties of any material do not decide its behavior around phase transition. Order parameter's dimension and degrees of freedom of the system decide it. Different phase transitions show same critical behavior around their transition temperatures, which is called as universality. To study critical phenomena scientist developed twin concepts are called as scaling and universality. At the near the critical point every systems shows self-similar properties in every scale, therefore they do not change under scale transformations at a point. This is a feature of the scaling. Universality word comes from different systems shows quite similar properties near critical point. In the statistical physics universality hypothesis is described as a number of macroscopic properties of a system are substantially independent from its microscopic structure in its universal class. Hence universality provides testable estimations without knowing details of microscopic structure. Universality classes are used to understand critical behaviors of model systems in it. If the order parameter works correct dimensions for a model, critical exponents will be same of the other systems in same universality class.       
In this work we investigated the Halperin-Lubensky-Ma (HLM) theory in 8CB – 10CB liquid crystal binary mixtures (8CB_(1-x) 10CB_x) using the birefringence measurements. Here x is the mole fraction of 10CB and x= 0.099, 0.179, 0.199, 0.300, 0.330, 0.400, 0.430, 0.499, 0.569, 0.629, 0.699, 0.749 ve 0.800 concentrations were studied. To prepare the concentrations, firstly liquid crystals amount were calculated and they weighted separately, then they were mixed and heated until they reached to isotropic phase and they were filled to liquid crystal cells to measure their birefringence. The birefringence corresponds to polarizing the incoming light into two light which have different speed and different refractive index. Anisotropic crystals shows birefringence. Liquid crystals are anisotropic materials, therefore they exhibit birefringence. To characterize the phase transitions different anisotropic properties can be used, one of them is birefringence. The light which left the liquid crystal carry internal order information of it. This order is determined via order parameter and it is showed with universal models. 
The birefringence measurements were performed by using a high-resolution rotating analyzer method. This method has resolution and high accuracy. Light comes from laser and passes through the firstly in sample which shows birefringence  property, light comes off as elliptical polarized and goes to quarter wave plate which turns it to plane polarized light. However in here there is a phase difference, direction of the polarization changed as θ. Phase difference, which corresponds to birefringence, is measured by lock-in amplifier. 
 If the light comes in an anisotropic material, it is polarized as two light inside it. This feature is known as birefringence and designated by ∆n. If the light is refracted perpendicular to optical axis this refractive index is known as ordinary refractive index n_o. If the light is refracted parallel to optical axis, this refractive index is named as extraordinary refractive index n_e. In the nematic phase the birefringence is defined as Δn=n_e-n_o. In the previous works nematic order parameter S(T) was related to brifringence data using S(T)=Δn(T)/〖Δn〗_0  relation in our work S(T) was found using birefringence data. After Anisimov et.al showed that if the coupling between SmA order parameter ψ and director flactuations is added to free energy there should be a cubic term. So this cubic term makes N – SmA transition first order. 
HLM theory is also saying that N – SmA transition is first order because of the fluctuations in the N – SmA phase transition. In the nematic phase continuous rotational symmetry of isotropic phase is broken and molecules have average orientation along the director. In smectic A phase layers parallel to director are formed, translational symmetry parallel to director is broken in this phase. Since layers' normal intends to be parallel locally, smectic order parameters will be change if director fluctuate as δn ̂. Therefore smectic order parameters and director field are in coupling strongly which makes N – SmA transition first order phase transition.  Anisimov et.al. showed that universal scaling function is given with S ̅/S ̅^* -(S ̅/S ̅^* )^(-1/2)=a ̂/S ̅^*  (x-x^* )≡y-y^* relation. Here x refers the concentration and S ̅ is the reduced entropy change. The relation between the nematic order parameter and S ̅ in the universal scaling function was related by Çetinkaya et.al. 
In this work the birefringence data was used to reveal the temperature behavior of the nematic order parameter. Phase diagram of mixtures is plotted from birefringence data and it was observed that there is only isotropic – smectic A phase transition for x≥0.630 mixtures. Therefore following calculations and analysis have been done for   x≤0.630 mixtures. Transition temperatures in the phase diagram are in good consistency with calorimetric measurements. 
To obtain order parameter birefringence data was fitted to a four-parameter fit equation and critical exponent, β, has been revealed and it is average value obtained as 0.245±0.002 which is consistent with tricritical nature of the nematic – isotropic phase transition.  
The order parameter S(T) obtained from birefringence data and the reduced entropy discontinuity S ̅ at N – SmA transition was extracted from S(T) data. We have found that  S ̅ values for the mixtures x≥0.3 is consistent with previous calorimetric studies. These S ̅ values were well fitted with the universal scaling function which is itself consistent with HLM theory. 
The temperature derivative of S(T) near N – SmA transition has been shown to exhibit the same power law divergence as the specific heat capacity with the critical exponent α. Then using this argument we have been obtained the effective critial exponent values of some binary mixtures in the vicinity of the N-SmA transition. For the mixture x=0.330, it was observed that the value of the effective critical exponent reaches its tricritical value α_TCP=0.5. Also for α values smaller than TCP almost zero latent heat has been observed. This is consistent beahivor with adiabatic scanning calorimetry measurements.