Saupe order parameters

Field et al.62 developed a general procedure to estimate the Saupe order parameters for spin systems aligned in liquid crystal media by iteratively fitting the experimental and simulated spectral widths of the high-order multiple quantum spectra. 

S and D are the Saupe order parameters that describe the order of the molecules in the absorbing sample. s% (i = 1, 2, 3) are the diagonal elements of the transition moment tensor. They are proportional to the squares of the components of the transition moments (<(/< > i = 1,2,3)) given with respect to a molecule fixed-coordinate system (xj) joc(Circular dichroism of chiral anisotropic phases without suprastructural chirality). 

The residual dipolar couplings have been extensively applied by Diirr et al in order to calculate Saupe order parameters in their studies on the alignment of three druglike compounds in lipid bilayers which have been carried out by solid-state F-NMR and molecular dynamics. 

H, Haller extrapolation (with fit exponent) MS, Mayer-Saupe order parameter P, HNMR order parameter data. 

Fig. 7 Temperature dependence of the Maier and Saupe order parameters of polymer DDA9-L and monomer PAA in pure melts (dashed lines), and in a 5 % PAA + 95% DDA9-L, by weight, mixture (DDA9-L, closed circles PAA, open...

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. 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

One of the characteristic features of mesogenic molecules in uniaxial mesophases is their orientational order specified by a set of order parameters, which forms a symmetric and traceless order matrix S. For rigid molecules, the elements of order matrix Sa/Y defined by Saupe,52 are given by... 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

Nematic phases are characterized by an unordered statistical distribution of the centers of gravity of molecules and the long range orientational order of the anisotropically shaped molecules. This orientational order can be described by the Hermans orientation function 44>, introduced for l.c. s as order parameter S by Maier and Saupe 12),... 

The electrostatic part, Wg(ft), can be evaluated with the reaction field model. The short-range term, i/r(Tl), could in principle be derived from the pair interactions between molecules [21-23], This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6,22], These are phenomenological models, essentially in the spirit of the popular Maier-Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 

Considering in this frame a certain transition moment parallel to the preferred molecular axis, it is evident that radiation with the electric vector oscillating parallel to the optical axis experiences stronger absorbance (Ay) than perpendicularly polarized radiation (absorbance A ). Closer inspection shows (Maier and Saupe, 1959) that Aj increases with increasing degree of order and thus, increasing order parameter S, while A decreases ... 

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. 

Figure 10.4 Measured values of the order parameter S — S2 versus temperature for MBBA using Raman scattering ([J), NMR (A), birefringence (A), and diamagnetic anisotropy (V), as described in Deloche et al. (1971). The solid line is the prediction of the Maier-Saupe theory the dashed line is a modification of the Maier-Saupe theory by Humphries, James, and Luckhurst (1972). (From de Gennes and Frost, Copyright 1993, by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)...

Problem 10.2 (Worked Example) Using the Maier-Saupe theory described in Section 2.2.2.2, derive an implicit equation relating the order parameter 5 as a function to the Maier-Saupe... 

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 2.15. The order parameter vs. reduced temperature for small molecular mass liquid crystals. (Modified from Maier Saupe, 1959.)...

It is worthwhile to point out that the Maier-Saupe theory has been successful in analyzing the behavior of small molecular mass liquid crystals at transition, such as the temperature change of the order parameter. The jump of the order parameter at transition, Sc = 0.43 is in reasonable agreement with most experiments. The Onsager and Flory theories, which take into account the steric effects predict a higher critical order parameter. 

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