Potential Maier-Saupe

We consider first the Maier-Saupe tlieory and its variants. In its original foniiulation, tills tlieory assumed tliat orientational order in nematic liquid crystals arises from long-range dispersion forces which are weakly anisotropic [60, 61 and 62]. However, it has been pointed out [63] tliat tlie fonii of tlie Maier-Saupe potential is equivalent to one in... 

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. 

Maier-Saupe potential 275 Markovian process adiabatic interference 145 asymptotics 38 impact theory 38 noise 227... 

The d, b n, and cmn coefficients are given numerically as polynomials of S0 and their expansion coefficients are tabulated in Table 1 of Voids for the Maier-Saupe potential. In the single exponential approximation for gmn(t),119 the spectral densities simplify to... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

An alternative nematic potential due to Maier and Saupe (1958, 1959, 1960) is perhaps more appropriate for thermotropic nematics. The Maier-Saupe potential is given by... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

Saupe potential, and U Uq = 10.19 for the Onsager potential. The two potentials are significantly different in their predictions of the dependence of the order parameter on U. For a given value of U/ U, the Onsager potential predicts a higher-order parameter than does the Maier-Saupe potential. 

Worked Example 10.5 of Chapter 10 shows how the Maier-Saupe potential can be used to predict the order parameter of a nematic. 

PHIC extrapolate to roughly 6.7, which is close to the value predicted by the Flory theory in the melt. This suggests that even for bulk HPC, the nematic-isotropic transition is driven primarily by excluded-volume, or packing, effects and only secondarily by anisotropic van der Waals interactions. The temperature dependence of the axial ratio could be incorporated into the Maier-Saupe potential by suitably adjusting the temperature dependence of the coefficient 17ms-... 

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. 

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. 

The Maier-Saupe potential is a phenomenological model originally proposed for thermotropic small-molecule LCs. It is obtained by replacing the excluded-volume interaction in Eq. (9) by... 

Split the Maier-Saupe potential into two terms and express the partition function as... 

Fig. 2.3.5. Orientational distribution functiony(cos ) as defined by (2.3.22) in the nematic phase of MBBA. Circles represent values from Raman measurements and the line gives the distribution function derived from a two-term Maier-Saupe potential in which the parameters are adjusted to give a good fit with the observed

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)). 

A similar analysis may be applied to the partially ordered nematic fluids composed of molecules comprising the mesogenic unit and flexible chain segments. In the LC state, one must consider the orientation-dependent interactions in addition to those of the isotropic nature. As mentioned earlier, the volume dependence (1/V ) incorporated in the Maier-Saupe expression may be replaced by MV. In its modified form, Maier-Saupe potential can easily be accommodated by introducing an additional term in the conventional van der Waals expression ... 

A closer look at this concentration dependence of the first positive maximum N, as well as the shear rate at which this occurs, was conducted by Baek et al [46]. Both were seen to be monotonically increasing functions up to concentrations of 40%. They demonstrated qualitative agreement with the predictions of Doi theory with the Hinch-Leal closure and the Maier-Saupe potential. They also note that the ratio of shear rate at which becomes negative to the shear rate of the first positive maximum remains constant at about 3.5. (Our data from 1978 and 1980 yielded an average ratio of 3.2 with a standard deviation of 0.9 for nine PBG solutions and an average ratio of 2.2 with a standard deviation of 0.3 for three PCBZL solutions.) They determined that the rapid increase in (dy/dt) with concentration cannot be attributed to a decrease in molecular relaxation time with... 

A detailed description of the molecular motions in the Maier-Saupe potential disturbed by the electric probe field has been done by Martin and co-workers. These authors obtained the numerical solutions for the relaxation times T and Tj and for polarization. They found that the relaxation process measured at nUE geometry is slowed down with respect to the Debye-type motion in the isotropic phase, whereas the second relaxation process connected with the molecular reorientations around the long axes (nlE geometry) becomes faster in the presence of the nematic potential, that is, gn > 1 and < 1. 

For cylindrical molecules in mesophases of Dooh symmetry, a truncated form of the pseudo-potential (Maier-Saupe potential) may be used,... 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

It was first noticed by Gelbart and Gelbart [16] that the predominant anisotropic interaction in nematics results from a coupling between the isotropic attraction and the anisotropic hard-core repulsion. This coupling is represented by the effective potential Veff(l, 2) = V(l, 2) Q(ji 2 z)- This potential can be averaged over all orientations of the intermolecular vector and then can be expanded in Legendre polynomials. The first term of the expansion has the same structure as the Maier-Saupe potential /(r,2)72((fli 02)) but with the coupling constant J determined... 

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