Nematic phase equilibria

Since a in the equilibrium nematic phase is fairly large (a >5 under usual conditions), Eq. (22) may be truncated at the third or fourth term. 

The equilibrium value of a in the nematic phase can be determined by minimizing AF. With Eq. (19) for AF from the scaled particle theory, S has been computed as a function of c, and the results are shown by the curves in Fig. 12. Here, the molecular parameters Lc and N were estimated from the viscosity average molecular weight Mv along with ML and q listed in Table 1, and d was chosen to be 1.40 nm (PBLG), 1.15 nm (PHIC), and 1.08 nm (PYPt), as in the comparison of the experimental phase boundary concentrations with the scaled particle theory (cf. Table 2). 

It has been found for some systems, and may be true for all, that there is no transition directly from the isotropic to the nematic phase as the critical condition is attained. Instead, a narrow biphasic region is found in which isotropic and nematic phases co-exist. This behaviour was predicted by Flory 2), even although his initial calculations related to monodisperse polymers. It is accentuated by polydispersity (see Flory s review in Vol. 59 of Advances in Polymer Science), and indeed for a polydisperse polymer the nematic phase is found to contain polymer at a higher concentration and of a higher average molecular weight than the isotropic phase with which it is in equilibrium. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Not only reaction rates, but also the positions of chemical equilibria can be influenced by liquid crystals as reaction media. A nice example is the ionization equilibrium of chloro-tris(4-methoxyphenyl)methane according to Ar3C-Cl Ar3C+ -I-C1 , which is more shifted in favour of the nearly planar triarylcarbenium ion in nematic liquid crystals as compared to in an isotropic reaction medium [868], Obviously, the discshaped carbenium ion fits better into the rod-Hke nematic phase than the tetrahedral covalent ionogen, which distorts the internal structure of the nematic liquid crystal. 

Sluckin adopted a quasi-equilibrium thermodynamic approach to understanding the effect of a strain rate field on the isotropic-anisotropic transition in polymer solutions. He derived a Clausius-Clapeyron-like equation which connects the shift in the critical polymer mole fraction C, and Cj, which are concentrations of isotropic and nematic phases, respectively, to the applied strain rate. 

While it was assumed above that only G. is affected by thermal history, in the case of main chain polymeric liquid crystals pronounced time dependent variability in G c has recently also been observed (7,8). It was shown that the lack of equilibrium perfection in the nematic phase can lead to substantial depression of the isotropization temperature T c =T. Thus non-equilibrium mesomorphic states can also, in principle, affect the phase sequence-(enantiotropic, monotropic) in the case of polymeric liquid crystals. 

Not only does the melting process depend on the thermal history of the sample, the isotropization of the polymeric nematic phase is also history-dependent. Feijoo and coworkers (1988 1990) are apparently the first to have studied in detail the effect on the isotropization of the thermal treatment. According to these researchers the establishment of thermodynamic equilibrium in the nematic phase was not as quick as has been commonly believed, for both the nematic and the isotropic phases are fluid states. They found that the heat and temperature of isotropization of the polymeric nematic phase were remarkably affected by the preceding... 

Since the liquid crystal forms the continuous phase of the binary mixture, we are only interested in a small part of the total phase diagram. Weight fractions of the liquid crystal in the range 0.9 to 1 were used to determine the partial phase diagram of the mixture which is shown in Fig. 2. The system forms an isotropic (I) phase at high temperature, and a diphasic equilibrium between an isotropic and a nematic phase (N-i-I) at low temperature. A nematic domain (N) is found at intermediate temperatures and low silicone oil concentrations. As pointed out in the experimental section, the existence of this nematic domain has some importance prior to quenching the system to the diphasic region. The present mixture exhibits classical features usually observed in other mixtures of nematic liquid crystals and polymers or isotropic fluids [29,30]. 

At the same point, the value of for the isotropic phase in equilibrium with the nematic phase is... 

Let us first consider the set of equidistant diffuse streaks (c) perpendicular to the director [la,b, 30]. These streaks are very similar to those observed on the X-ray scattering patterns of the nematic phases of main-chain LCPs discussed in Sect. 3.2 but they must be interpreted differently because the SmA phase shows (quasi) long-range positional order. These diffuse streaks also correspond to the intersection with the Ewald sphere of a set of equidistant diffuse planes. But here this set represents the Fourier transform of uncorrelated rows of side-chains displaced along the director from their equilibrium position inside the layers (Fig. 13). 

Energy parameters and charge-transfer spectra of complexes of Br2 with several substituted pyridines have now been compared with the force constants k(Br—Br) and k(N- Br2) and with the properties of the donors. The complexes with orr/to-substituted pyridines show systematic deviations from the relations found to be valid for the other donors. The n.m.r. spectrum of the pyridine-Br2 complex in a nematic phase has been obtained and analysed. The results indicate that the donor-acceptor interaction is similar to that found in the solid state for other halogen-pyridine complexes. The equilibrium constant for the formation of the 1 1 complex of Br2 with hexamethyl-phosphortriamide (HMPA) has been determined by n.m.r. spectroscopy. Solid adducts of Br2 with poly-HMPA could also be prepared. 

Without mechanical stress (a=0), spontaneous deformation must occur in the nematic phase because of the linear coupling between S and e. Minimizing F with respect to e gives the equilibrium value of = USIE. Inserting this value and the form for Fq, the above equation takes the follow-... 

Studies on the sorption of some hydrocarbons have shown that above the transition temperature of EBBA (331 K) the isotherms obey Henry s law and the solubility coefficients S can be calculated. The sorption and desorption curves are similar in shape which indicates that these systems follow Fickian sorption. This fact indicates that steady state surface equilibrium is reached and that the diffusion coefficient for hydrocarbons is a function of concentration only. It follows that the membranes containing 60 wt.% of EBBA are homogeneous from the view point of gas permeation at the temperature above transition in EBBA. The permeability coefficients P show a distinct jump in the vicinity of transition temperature from crystal to nematic phase. This phenomenon was observed for hydrocarbon gases, noble gases like He, and for inert gases like N2. 

Flexible chains, which tend to form a random coil in the isotropic state, are highly incompatible with LCs. Phase equilibrium studies indicate that chain molecules such asn-alkanes [Kronbergetal., 1976 Orendi and Ballauff, 1989], dimethyl ethylene glycol ethers [Abe et al., 1993], poly(oxyethylene) [Dubault et al., 1982], or polystyrene [Ballauff, 1986] are very disruptive to the nematic order and thus only a limited amount is allowed to exist in LCs [Martire, 1979]. 

It is very interesting to see that a crossover between the equilibrium flexodomains and the dissipative EC patterns can be observed in the same experiment (planar geometry) by merely increasing the AC frequency co. Inspection of Fig. 4.5 reveals that at very small u) and for suitable material parameter combinations of the nematics, the critical voltages of the equilibrium flexodomains and those of the dissipative electroconvection patterns might approach each other. In fact, very recent theoretical and experimental studies on the calamitic nematic Phase 4, have demonstrated the... 

As in low molecular systems, the nematic phase exhibits orientation order between rod-like units, but no positional order. Due to orientation of mesogenic units also spacers are oriented by stretching. Molecules are oriented with respect to director. Thus, the isotropic - nematic transition is a disorder-order transition from a phase without long range order to a phase with long range orientation order but still liquid-like. It is an equilibrium transition since there is no undercooling. This transition is characterized by = S -S <0 and < 0. We may relate it to equation of Clausius-Clapeyron. It follows ... 

Equations (10.75) and (10.78) determine the stress for a given velodty gradient and can be regarded as a constitutive equation. It should be emphasized that this constitutive equation holds in both the isotropic and the nematic phases since no presumption has been required about the equilibrium state. 

As mentioned above, the experimentally observed Tm will only allow the difference in the regular solution parameters to be calculated but to calculate the phase diagram the individual parameters have to be known. For simple minimum or maximum diagrams where two non-ideal liquid phases appear, for example, a nematic-isotropic equilibrium, two regular solution parameters are necessary. To obtain the individual values once knowing... 

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