Molecular Theory of the Nematic Phase

Goossens WJA. 1971. Molecular theory of the cholesteric phase and of the twisting power of optically active molecules in a nematic liquid crystal. Mol Cryst Liq Cryst 12(3) 237 244. 

We start with the microscopic definitions and discussion of the nematic and smectic order parameters and then proceed with some elementary information about anisotropic intermolecular interactions in liquid crystals. Then we discuss in more detail the main molecular theories of the nematic-isotropic phase transition and conclude with a consideration of molecular models for smectic A and smectic C phases. 

Lopatina and Selinger recently presented a theory for the statistical mechanics of ferroelectric nanoparticles in liquid crystals, which explicitly shows that the presence of such nanoparticles not only increases the sensitivity to applied electric fields in the isotropic liquid phase (maybe also a possible explanation for lower values for in the nematic phase) but also 7 N/Iso [327]. Another computational study also supported many of the experimentally observed effects. Using molecular dynamics simulations, Pereira et al. concluded that interactions between permanent dipoles of the ferroelectric nanoparticles and liquid crystals are not sufficient to produce the experimentally found shift in 7 N/ so and that additional long-range interactions between field-induced dipoles of nematic liquid crystal molecules are required for such stabilization of the nematic phase [328]. 

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

Here co is frequency of oscillating electron in the first atom, Kq is polarizability of the second atom, r is distance between the two. This result is in qualitative agreement with a quantum-mechanical theory developed by London. For calculation of parameter A in (6.71) Maier and Saupe used this basic formula, but in addition they took the anisotropy of molecular polarizability Act into account. It is Aa that determines the stability of the nematic phase. 

Ma Y, Zha L, Hu W, Reiter G, Han CC (2008) Crystal nucleation enhanced at the diffuse interface of immiscible polymer blends. Phys Rev E 77(6) 061801 Maier W, Saupe A (1959) A simple molecular statistical theory of the nematic crystalline-liquid phase. Zeitschrift fur Naturforschung 14 882-889 Mandelkem L (1964) Crystallization of polymers. McGraw-Hill, New York Matsuoka S (1962) The effect of pressure and temperature on the speciflc volume of polyethylene. J Polym Sci 57(165) 569-588... 

Maier and Saupe [13] developed a statistical theory to describe the liquid crystalline state and the molecular ordering for the nematic phase. In analogy to the treatment of ordering phenomena in ferromagnetics or ferroelectrics, this theory describes the intermolecular orientational forces by a mean field method. Each individual molecule feels a nematic potential D = f (0, S, V) which depends on the momentaneous angle 6 between its long axis and the optic axis, the order parameter S and the molar volume V. S is then given by... 

The interaction potentials (Eqs. 85 and 90) essentially depend on a coupling between the molecular orientation and the intermolecular vector. We note that this coupling could be neglected in the first approximation in the theory of the nematic-smectic A transition, as it is done, for example, in the McMillan theory. At the same time this coupling just determines the effect in the theory of transition into the smectic C phase. 

Molecular-statistical theories are available for several different smectic phase types [25]. In addition to the ingredients of the nematic phases, the theories incorporate parameters responsible for the formation of layers. The clearing temperatures, however. 

In this chapter we consider the very simplest approach to the molecular theory of liquid crystals. We shall approach the theory phenomenologically, treating the problem of the existence of the nematic phase as an order-disorder phenomenon. Using the observed symmetry of the nematic phase we shall identify an order parameter and then attempt to find an expression for the orientational potential energy of a molecule in the nematic liquid in terms of this order parameter. Such an expression is easily found in the mean field approximation. Once this is accomplished, expressions for the orientational molecular distribution function are derived and the thermodynamic functions simply calculated. The character of the transformation from nematic liquid crystal to isotropic fluid is then revealed by the theory, and the nature of the fluctuations near the transition temperature can be explored. 

A rigorous molecular theory of a fluid system based on a pairwise interaction potential as complicated as Eq. [2] is impossibly difficult. A simple but adequate approach is to derive a theory in the mean field approximation. That is, we derive a single molecule potential that serves to orient the molecule along the symmetry axis (the director) of the nematic phase. The single-molecule potential represents (approximately) the mean field of intermolecular forces acting on a given molecule. Mean field theories have been found capable of describing the qualitative behavior of many different cooperative phenomena. The... 

Maier W, Saupe A. A simple molecular statistical theory of the nematic crystalline-liquid phase. Z Naturf 1959 14 882-889. 

The effect of anisotropic interactions, orientation-dependent interactions in particular, which is responsible for the stability of the nematic phase to some degree, is prevalent in all mixtures. This question has been assigned an important place in the theory of low-molecular-weight liquid crystals of Melw and Saupe [49]. In the review of Flory s woric in [30], it was emphasized that although orientation-dependent interactions in polymers containing phenylene units, for example, can cause stabilization of the liquid-crystalline state, the asymmetry of the molecular shape is undoubtedly the dominant molecular characteristic responsible for the liquid-crystalline state in such systems. 

XL13654, and SCE9. A well-studied ferroelectric liquid crystal is DOBAMBC its molecular structure is shown in Figure 4.1 Ic. Because of these differences in the degree of order and molecular arrangement and the presence of a permanent dipole moment, the physical properties of smectic liquid crystals are quite different from those of the nematic phase. In this and the following sections we examine the pertinent physical theories and the optical properties of three exemplary types of smectics smectic-A, smec-tic-C, and (ferroelectric) smectic-C. ... 

Maier, W, and A. Saupe. 1959. A simple molecular-statistics theory of the nematic liquid-crystaUine phase, part 1. Z. Natuiforsch. 14a 882-900. 

Currently, theories are not yet able to predict the transition temperatures based on molecular structure of the constituent molecules. However, for several compounds there is considerable empirical data relating the transition temperature between isotropic and nematic phases (Tni) to molecular structure. Higher implies greater nematic stability. For example, it is... 

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 the merit of Picken (1989, 1990) having modified the Maier-Saupe mean field theory successfully for application to LCPs. He derived the stability of the nematic mesophase from an anisotropic potential, thereby making use of a coupling constant that determines the strength of the orientation potential. He also incorporated influences of concentration and molecular weight in the Maier-Saupe model. Moreover, he used Ciferri s equation to take into account the temperature dependence of the persistence length. In this way he found a relationship between clearing temperature (i.e. the temperature of transition from the nematic to the isotropic phase) and concentration ... 

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