Order parameters, thermotropic liquid crystals

We note that earlier research focused on the similarities of defect interaction and their motion in block copolymers and thermotropic nematics or smectics [181, 182], Thermotropic liquid crystals, however, are one-component homogeneous systems and are characterized by a non-conserved orientational order parameter. In contrast, in block copolymers the local concentration difference between two components is essentially conserved. In this respect, the microphase-separated structures in block copolymers are anticipated to have close similarities to lyotropic systems, which are composed of a polar medium (water) and a non-polar medium (surfactant structure). The phases of the lyotropic systems (such as lamella, cylinder, or micellar phases) are determined by the surfactant concentration. Similarly to lyotropic phases, the morphology in block copolymers is ascertained by the volume fraction of the components and their interaction. Therefore, in lyotropic systems and in block copolymers, the dynamics and annihilation of structural defects require a change in the local concentration difference between components as well as a change in the orientational order. Consequently, if single defect transformations could be monitored in real time and space, block copolymers could be considered as suitable model systems for studying transport mechanisms and phase transitions in 2D fluid materials such as membranes [183], lyotropic liquid crystals [184], and microemulsions [185],... 

Thermotropic liquid crystal phases are formed by rodlike or disclike molecules. However, in the following we consider orientational ordering of rodlike molecules for definiteness, although the same parameters can be used for discotics. In a liquid crystal phase, the anisotropic molecules tend to point along the same direction. This is known as the director, which is a unit vector denoted n. 

Structure of thermotropic liquid crystals is rather well understood. There are three main structural types nematic, cholesteric, and smectic. In nematic liquid crystals molecules are aligned approximately in the same direction, but positionally molecules are disordered. An axis of preferable molecular orientation is called a director. More precisely, the director is defined as a unit vector n(r) that is parallel to the molecular orientation at the point r. If we use the long axis of the molecules as a reference and denote it as k, the microscopic scalar order parameter 5 is defined [16,17] as follow ... 

Here we give an overview of fluctuation dominated thermotropic liquid crystal phase transitions with a few hints of emerging aspects. From this perspective, the situation may be fairly summarized by noting that while analogies to phase transition models in spin-space (e.g. XY model with two components for the order parameter or Ising model with one) or momentum space (superconductivity) are a powerful tool to predict qualitative behavior for fluctuation dominated, real space, high temperature liquid crystal phase transitions, there is a significant gap between several quantitative (and qualitative) expectations and experimental measurements. 

The films may be stacked in layers. There are second order parameters describing the molecular array more specifically. The basic smectic arrangement can be found in both thermotropic and lyotropic liquid crystals. 

The LC phases, illustrated by examples in Figures 2 to 4, originate from different driving forces. In the first example, heat transfer causes transition from isotropic to LC state. The order parameter in the second example is ruled by concentration of MBBA in solution. In the last example the mesophase is formed by light exposure of the material. We see, liquid crystals develop either under thermotropic or lyotropic conditions depending on the driving force for organization of the system. 

Figures 9 and iO show the results from one experiment on PPC deep in the thermotropic phase at 65°C. A steady state shear flow of 2.5s is applied and the level of global preferred orientation is evaluated from a series of x-ray scattering patterns recorded with a cycle time of 15s over a period of 1000s. Using the azimuthual variation of intensity of the diffuse peak at IQI 0.4 A , values of the global orientation parameters mtd are obtained as a function of time. A value of , = 0 indicates a completely isotr opic state, while . = I indicates a fully aligned system. A nematic liquid crystal systems exhibits microscopic order...

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