Liquid crystals lattice theory

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

The diamter of the PBG a-helix has been estimated variously to be between 15-25 A as noted above (25). Values of d used in the determination of persistence len s are listed in the solvent categories of Table I. Values of Vp calculated from Equation 1 using x = L/d are invariably too low for the solvents of this study. Better agreement with the lattice theory has been reported for critical volume fractions of PBG in dimethylformamide and m-cresol (32). Experimental volume fractions for liquid crystal formation of PM in dioxane are lower, however, than those calculated from Flory s theory (33). PBG is known to undergo extensive... 

The question whether the dispersion bands observed with feebly damped waves by Colley, Obolensky, Romanoff, Potapenko, and others (Handbuch der Physiky 15, 514 et seq. (Berlin, 1927)) in the region of wave-lengths of a few decimetres correspond to intramolecular vibrations might be determined by comparison of the spectra of the vapours with those of the liquids. As is well known, the kinetic theory of liquids has recently exhibited a decided tendency to follow the theory of crystal lattices more closely in many respects for a comprehensive account of some of the most important papers on this subject see K. Jellinek, Lehrbuch der physikalischen ChemiCy 1, 824 et seq, especially pp. 828, 831 (Stuttgart, 1928). 

Polarized analysis There is useful spectral information arising from the analysis of polarization of Raman scattered light. This, typically called as polarized analysis, provides an insight into molecular orientation, molecular shape, and vibrational symmetry. One can also calculate the depolarization ratio. Overall, this technique enables correlation between group theory, symmetry, Raman activity, and peaks in the corresponding Raman spectra. It has been applied successful for solving problems in synthetic chemistry understanding macromolecular orientation in crystal lattices, liquid crystals or polymer samples and in polymorph analysis. 

Brownian motion kinetic theory ideal gas pressure crystal lattice liquid crystal amorphous material plasma... 

A weakness of this simple theory is the assumption that once crystal forming material arrives at the liquid/crystal interface it is immediately incorporated into the existing crystal lattice. It is apparent however, that a finite time is likely to be required to enable this process to be carried out, i.e. there is a resistance to the assimilation of new material. This resistance may be expressed in terms of a "rate of reaction" for the solute molecules to arrange themselves into a crystal lattice. If the concentration of solute in solution at the crystal surface/solution interface is c,.. Equation 8.5 for diffusion may be written as ... 

Are all quantitative predictions of the thermodynamics of liquid crystals correct. If not stop here. The reason for this step is that die theory (Flory-Huggins lattice model) also predicts the occurrence of the isotropic to nematic phase transition in liquid crystals. If the theory had predicted correctly the properties of glasses but had failed for liquid crystals we would have had to abandon it, especially since in both cases the cause of the transition is ascribed to the vanishing of the configurational entropy. Alternatively the correctness of the prediction for liquid crystals argues for the correctness of the prediction for glasses. Since we have not been stopped by steps 3 and 4 we proceed to step 5. 

SINCE the discovery of liquid crystalline phenomenon for low molecular weight liquid crystals (LMWLCs) more than 100 years ago, anisotropic ordering behaviors of liquid crystals (LCs) have been of considerable interest to academe [1-8], In the 1950s, Hory postulated the lattice model for various problems in LC systems and theoretically predicted the liquid crystallinity for certain polymers [1-3], As predicted by the Hory theory, DuPont scientists synthesized lyotropic LCPs made of rigid wholly aromatic polyamide. Later, Amoco, Eastman-Kodak, and Celanese commercialized a series of thermotropic main-chain LCPs [2]. Thermotropic LCPs have a unique combination of properties from both liquid crystalline and conventional thermoplastic states, such as melt processibility, high mechanical properties, low moisture take-up, and excellent thermal and chemical resistance. Aromatic main-chain LCPs are the most important class of thermotropic LCPs developed for structural applications [2,4-7]. Because they have wide applications in high value-added electronics and composites, both academia and industry have carried out comprehensive research and development. 

Murthy CS, Singer K, Klein ML et al (1983) Electrostatic interactions in molecular crystals. Lattice dynamics of solid nitrogcm and carbon dioxide. Mol Phys 50 531-541 Hirschfelder JO, Curtiss CF, Bird RB (1954) Molecular theory of gases and liquids. Wiley, New York... 

The heat capacity and transition enthalpy data required to evaluate Sm T ) using Eq. 6.2.2 come from calorimetry. The calorimeter can be cooled to about 10 K with liquid hydrogen, but it is difficult to make measurements below this temperature. Statistical mechanical theory may be used to approximate the part of the integral in Eq. 6.2.2 between zero kelvins and the lowest temperature at which a value of Cp,m can be measured. The appropriate formula for nonmagnetic nonmetals comes from the Debye theory for the lattice vibration of a monatomic crystal. This theory predicts that at low temperatures (from 0 K to about 30 K), the molar heat capacity at constant volume is proportional to Cv,m = aT, ... 

Recently, phase behaviour of mixtures consisting of a polydisperse polymer (polystyrene) and nematic liquid crystals (p-ethoxy-benzylidene-p-n-butylani-line) was calculated and determined experimentally. The former used a semi-empirical model based on the extended Flory-Huggins model in the framework of continuous thermodynamics and predicted the nematic-isotropic transition. The model was improved with a modified double-lattice model including Maier-Saupe theory for anisotropic ordering and able to describe isotropic mixing. ... 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

In Fig. 4.10, one can clearly see that the L-S curve is mainly determined by E Ec, and the L-L curve is mainly controlled by B/ (. but is also slightly affected by EpIEc- By changing the values of Ep/E and BjE, the interplay between crystallization and liquid-liquid demixing can be studied in a combination of the lattice theory and parallel molecular simulations (Hu and Frenkel 2004 Ma et al. 2007, 2008), which has been introduced in the author s book (Hu 2013). 

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