Maier-Meier theory

For comparison, the results obtained using the Maier-Meier theory [4] are also shown this is a generalization of the Onsager model [13] to uniaxial media. The same dipole moment used for the calculations with the molecular shaped cavity was assumed, and the radius a was taken to be 3.9 A, a value derived from the density of the system. Improvement of the predictions, when the sphere is replaced by a molecular shaped... 

Figure 2.33 Dielectric permittivity of 5CB. Experimental data [2] (solid line), and theoretical results obtained with the IEF method (filled diamonds) and with the Maier-Meier theory [4] (open diamonds).

In experiment on nematic liquid crystals, both positive and negative anisotropy <, is observed, the sign depending on chemical structure. The magnitude of e is often proportional to orientational order parameter S. In the isotropic phase the anisotropy disappears. Typical temperature dependencies of n and , are shown in Fig. 7.5. These observations can be accounted for by the Maier-Meier theory [5]. The latter is based on the following seven assumptions ... 

Fig. 7.7 Location of a molecular dipole moment with respect to the longitudinal molecular axis of a molecule. Note that in the Maier-Meier theory the dipole moment forms angle P with the axis of maximum polarizability of a spherical molecule...

This effect originates from the anisotropic dipole-dipole correlations not accounted for by the Maier-Meier theory operating with a single particle distribution function. When, with decreasing temperature, the smectic density wave p(z) develops (even at the short-range scale) the longitudinal dipole moments prefer to form antiparallel pairs and the apparent molecular dipole moment becomes smaller. This would reduce positive s. Theoretically, dipole-dipole correlations may be taken into account by introducing the so-called Kirkwood factors. 

Recently, Sharma has proposed some extension of the Maier-Meier approach to the case of nematogens with antiparallel dipole-dipole correlations of the molecules. He treated a polar LC material as a mixture of unpaired molecules with a finite dipole moment /u. and antiparallel pairs with zero dipole moment. The molecules interact with each other through a combination of the generalized Maier-Saupe pseudopotential for nematic mixtures and a reaction field energy term calculated from an extension of the Maier-Meier theory. Additionally, it was assumed that a dipole with dipole moment fi is embedded in a spherical cavity of dielectric permittivity n, which is surrounded by a medium of average dielectric permittivity e. In that case the expressions for the cavity field factor h and the reaction field factor / are given by h + n ), /= (e - rt")/[2rre a (2e-i-n )] and the left sides of... 

EQNS (1), (2) and (3) are commonly used as the basis of the molecular interpretation of the static permittivities measured as a function of temperature (e.g. [6,12-15,20,21,28-30,36,37]) and/or pressure [30,38]. Recently, quite successful predictions of anisotropic optical and dielectric constants from molecular modelling calculations were achieved with the aid of the Maier-Meier theory [39,40]. It seems worthwhile, therefore, to analyse these equations in order to point out the weak and strong points of the theory. EQN (3) is the most convenient for this purpose (both components of the permittivity are discussed by Jadzyn et al in a recent paper [41]). The parameters N, F and h in the Maier-Meier equations vary little with temperature. Therefore, the contribution from the polarisability anisotropy Aa to A8 varies with temperature in the same way as the order parameter S, whereas that connected with the orientation polarisation varies like S/T. Especially interesting seems to be the case of constant temperature discussed in [38] where As was measured as a function of pressure, p. The discussion of the measured permittivities, Sj and the anisotropy As as a function of the order parameter S obtained from the independent experiment seems to be the best way of verifying the assumptions on which the theory is based. 

Maier function, hard-core repulsion 56 Maier-Meier equations 236 f Maier-Saupe theory 44 f, 279 f, 332... 

For such a comparison one has to consider the step of the density at the phase transition. All the other relations which connect molecular parameters such as the molecular dipole moment /i, the polarizability a and the angle between the molecular long axis and fj, with each other have a general problem the calculation of the internal field and its anisotropy. Therefore, all the equations given in Vol. 1, Chap. VII.2 are necessary and useful but one has to take always into account the limitations of the models. Nevertheless, the Onsager theory [27] (basis for the Maier-Meier model [28]) and Kirkwood-Frbhlich model [29] have been... 

The dielectric anisotropy Ae of LC-materi-als is defined by Ae= j -ex, where and ej. are the dielectric constants parallel and perpendicular to the director. From the Maier and Meier theory it can be seen that both the polarizability anisotropy Aa and the permanent dipole movement /i of the LC molecule determine the dielectric anisotropy... 

In contrast to diamagnetism where the magnetic interactions between molecules can be neglected, the polarization field in the medium becomes important when discussing dielectric anisotropy (see, e.g., Bottcher< >). Maier and Meier > took this into account by applying the Onsager theory.The effective induced dipole moments per molecule along and... 

Dielectric constants of a liquid crystal are mainly determined by the dipole moment (/r), its orientation angle ff) with respect to the principal molecular axis, and order parameter (5), as described by the Maier and Meier mean field theory [27]... 

Maier and Meier [35] worked out these ideas more quantitatively by applying Onsager s theory of static polarization for nematic liquid crystals. The Onsager theory relates the dielectric constant to molecular properties, namely the molecular polarizability a and the permanent electrical dipole moment /i. In the case of nematic liquid crystals, however, the polarizability has to be treated as a tensor with principal values an and a and an angle ]8 has to be introduced to describe the... 

In the Onsager theory of isotropic dielectrics as well as in its extension to the nematic phase given by Maier and Meier the short range dipole-dipole correlations w e ignored. Therefore the dipole moment p in EQNS (1) - (3) cannot be identified with its value measured in the state [16]. The dipole-dipole correlations were considered in the theory developed by Frdtdich [17] who generalised the former Kirkwood approach [18]. Frdhlich has introduced the dipole-dipole correlation factor (known as the Frdhlich-Kirkwood g-factor) in the form... 

The dielectric theory for such a material has been given by Attard and is a development of the theories of Nordio et aV and Maier and Meier for the dielectric permittivity of a monodomain of LC material to include the case of partially aligned material. This theory involves the Wigner rotation matrix method for coordinate transformations. A simpler treatment has been given recently by Attard, Araki and Williams and uses a cartesian rotation matrix method for coordinate transformations. Maier and Meiershowed that the principal static permittivities for a monodomain of a uniaxial liquid crystal are given by... 

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