Scalar Orientational Order Parameter

In contrast to the bond-orientational order parameters mentioned above, scalar measures for translational order [that is, of the tendency of particles (atoms, molecules) to adopt preferential pair distances in space] have not been well studied. However, a number of simple metrics have been introduced recently (Truskett et al., 2000 Torquato et al., 2000, Errington and Debenedetti, 2001) to capture the degree of spatial ordering in a many-body system. In particular, the structural order parameter t. 

The polar orientational order parameter can be introduced as the scalar product of the polar director L and the normal to the surface h... 

For a uniaxial nematic phase, Qy is simply replaced by the scalar S, the nematic ordo param although tha e are many orientational order parameters S is normally dominant. For a uniaxial nematic phase, the constant wder parameter approximation leads to ... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

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-... 

Ginzburg-Landau models that in addition to the scalar order parameter for the concentrations of oil, water, and amphiphile contain a vector order parameter for the amphiphile orientation have also been studied [41, 45, 49-53]. 

Structure of thermotropic liquid crystals is rather well understood. There are three main structural types nematic, cholesteric, and smectic. In nematic liquid crystals molecules are aligned approximately in the same direction, but positionally molecules are disordered. An axis of preferable molecular orientation is called a director. More precisely, the director is defined as a unit vector n(r) that is parallel to the molecular orientation at the point r. If we use the long axis of the molecules as a reference and denote it as k, the microscopic scalar order parameter 5 is defined [16,17] as follow ... 

The only difference between the nematic phase and the isotropic phase is the orientational order. A proper description of this orientational order requires the introduction of a tensor of the second rank [7, 8]. This tensor can be diagonalized and for anisotropic liquids with uniaxial symmetry, the nematic phase can be described by only one scalar order parameter. The thermodynamic behavior in the vicinity of the N-I transition is usually described in terms of the mean-field Landau-de Gennes theory [7]. For the uniaxial nematic phase one can obtain the expansion of the free energy G in terms of the modulus of an order parameter Q. 

The same study has been done for a three-dimensional case. The peculiarity of the three-dimensional state makes one scalar order parameter not enough to describe the degree of ordering. That is why we used the shares of monomers rii = Ni/N, oriented along directions x, y and z (LNi = N, i = x, y, z) as order parameters for these values the normalization requirement is valid nx + ny + n += 1. 

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