Nematic potentials equation

Here Wgn are the coefficients of the expansion of the electrostatic free energy, which can be obtained from the free energy Wfl(li), according to Equation (2.269). T2n are the irreducible spherical components of the (second rank) surface tensor, which describe the anisometry of the molecular shape, and can be calculated in the form of integrals over the molecular surface [25]. Given the nematic potential the distribution function... 

The Doi theory captures the molecular viscoelasticity of LCP, i.e., the relaxation of the orientation distribution under flow. But it completely ignores distortional elasticity and is limited to monodomains. The assumption of spatial uniformity underlies all its key elements the nematic potential, the kinetic equation, and the elastic stress tensor. Therefore, its successes are restricted to situations where distortional elasticity is insignificant. 

Having retardation factors, one can calculate the nematic potential q using equations derived originally by Meier and Saupe [Eq. (26)] or by Coffey and co-workers [Eq. (27), Fig. 3j. With the aid of Eq. (26) the nematic potential has been calculated as a function of temperature and pressure for various nematics. " For nCB compounds the q values vary between 3kJ/mol near T, point and 6kJ/mol near the... 

Note that the contribution to the stress from the nematic potential is independent of the deformation rate and is therefore elastic. The coefficients i are known as the Leslie viscosities. (The factors of V2 in the equations do not appear in the original literature because of different definitions of D and 52.) The Onsager reciprocal relations from irreversible thermodynamics require that 2 + 3 = 6 - 5. Conservation of angular momentum must also be satisfied by the director, which takes the form... 

The solutions to the anisotropic diffusion equation can be written as a series expansion, each term of which can be associated with a particular relaxation time. For a harmonic perturbation of the rotational distribution function, as occurs in a dielectric relaxation experiment with an ac electric field, it was found that a single relaxation time was sufficient to describe the relaxation of p, and this could be expressed in terms of the relaxation time Xq) for in the absence of a nematic potential by ... 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

U(0) is the orientation dependent local potential energy and U (6) accounts for any anisotropy due to the local environment. For instance, if the sample were liquid crystalline, U (O) and U (rc) would represent the potential well associated with nematic or smectic director. In addition to local potentials it is clear that f(0) depends on the experimentally controllable quantities p, E, and T. The solution to equation 16 is the series expansion referred to as the third-order Langevin function L,3(p)... 

The nematic mean-field U, the molecule-field interaction potential, WE, and the induced dipole moment, ju d, are evaluated at different orientations using Equation (2.263), and then the coefficients of their expansion on a basis of Wigner rotation matrices can be calculated, according to Equation (2.268). The permittivity is obtained by a self-consistency procedure, because the energy WE and the induced dipole moment / md, as well as the reaction field contribution to the nematic distribution function p( l), themselves depend on the dielectric permittivity. 

Our calculations show that to achieve good accuracy with Eqs. (4.233) and (4.234) in a wide range of temperature and frequency variations, it is necessary to retain at least five (odd k= 1, 3,..., 9) lower modes of the spectrum. We remark that the first three relaxational modes have once been evaluated both numerically [109] and analytically [82] in studies of dielectric relaxation in nematic liquid crystals, where the forms of the potential and of the basic equation coincide with those given by our Eqs. (4.224) and (4.225), respectively. 

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 ... 

If Eq. (11-3) is multiplied by uu and integrated over the unit sphere, one obtains an evolution equation for the second moment tensor S (Doi 1980 Doi and Edwards 1986). In this evolution equation, the fourth moment tensor (uuuu) appears, but no higher moments, if one uses the Maier-Saupe potential to describe the nematic interactions. Doi suggested using a closure approximation, in which (uuuu) is replaced by (uu) (uu), thereby yielding a closed-form equation for S, namely. 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

Nearly exact numerical solutions of the Smoluchowski equation show that for the Maier-Saupe potential, A < 1 when S = S2 > 0.524. For the Onsager potential, A < 1 for all values of the order parameter within the nematic range. Values of A for the Onsager potential are plotted in Fig. 11-18. 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features. There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential.However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). 

Although it is desirable to be able to capture the full dynamics of both ion transport and solvent transport [Asaka and Oguro (2000) Tadokoro et al. (2000)], such an attempt typically does not lead to analytical solutions. Therefore, we follow Nemat-Nasser and Li (2000) and focus on the dynamics of cations only. Let D, E, , and p denote the electric displacement, electric field, electric potential, and charge density, respectively. The following field equations hold ... 

Here, the first term describes the nematic-like elastic energy in raie crmstant approximation (K in 9). This term allows a discussion of distortions below the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a finite size). In fact, the most important specific properties of the antiferroelectric are taken into account by the interaction potential W between molecules in neighbour layers the second term in the equation corresponds to interaction of only the nearest layers (/) and (/ + 1). Let count layers from the top of our sketch (a) then for odd layers i, i + 2, etc. the director azimuth is 0, and for even layers / + 1, / + 3, etc. the director azimuth is n. The third term describes interactimi of the external field with the layer polarization Pq of the layer / as in the case of ferroelectric cells. Although for substances with high Pq the dielectric anisotropy can be neglected, the quadratic-in-field effects are implicitly accounted for by the highest order terms proportiOTial to P. ... 

The Maier-Saupe theory is extremely useful in understanding the spontaneous long-range orientational order and the related properties of the nematic phase. The single-molecule potential Vi(cos0) is given by Eq. (3.19) with e being volume dependent and independent of pressure and temperature. The self-consistency equation for (P2) is... 

We now use the Debye equation to treat molecular reorientation in a nematic liquid crystal [7.1, 7.2] by suitably modifying the equation to allow for an ordering potential that is known to be present in the mesophase. The orientational distribution function f Cl) and the conditional probability function f Qo n,t) are obtained by solving... 

Equation (27) presents a simple anisotropic attraction potential that favors nematic ordering. This potential has been used in the original Maier-Saupe theory [11, 12]. We note that the interaction energy (Eq. 27) is proportional to the anisotropy of the molecular polarizability Aa. Thus, this anisotropic interaction is expected to be very weak for molecules with low dielectric anisotropy. Such molecules, therefore, are not supposed to form the nematic phase. This conclusion, however, is in conflict with experimental results. Indeed, there exist a number of materials (for example, cyclo-... 

If one neglects the asymmetry of the molecular shape (i.e. puts the function 2=D= const in the effective potential) and uses the dipole-dipole dispersion interaction potential (Eq. 27), one arrives at the Maier-Saupe theory. In this theory the interaction potential contains only the P2((< i "2)) and as a result it is possible to obtain the closed equation for the nematic order parameter S. Substituting the potential V(l, 2)=-7(ri2) P2 di a2)) ry2 D) into Eqs. (27-29), multiplying both sides of Eq. (27) by P2((fli 02)) nd integrating over U2, we obtain the equation... 

This general equation applies both to nematic and smectic phases because the one-particle density may depend on position. In the case of a uniform nematic phase without an external potential we obtain... 

For rigid molecules the frequency dependence of the orientational polarization in isotropic liquids can be calculated using Debye s model for rotational diffusion. This may be modified to describe rotational diffusion in a liquid crystal potential of appropriate symmetry, but the resulting equation is no longer soluble in closed form. Martin, Meier and Saupe [34] obtained numerical solutions for a nematic pseudopotential of the form ... 

In the study of dielectric relaxation, temperature is an important variable, and it is observed that relaxation times decrease as the temperature increases. In Debye s model for the rotational diffusion of dipoles, the temperature dependence of the relaxation is determined by the diffusion constant or microscopic viscosity. For liquid crystals the nematic ordering potential contributes to rotational relaxation, and the temperature dependence of the order parameter influences the retardation factors. If rotational diffusion is an activated process, then it is appropriate to use an Arrhenius equation for the relaxation times ... 

According to an equation of Hiwatari and Matsuda [115] the p-V-T data for the nematic state yields an l/r ° dependence of the repulsive potential energy on the inter-molecular distance r or p , when the density p is used. McColl [116] had found p for 4,4 -bis-methoxy-azoxybenzene. 

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