Order parameter tensor

Eqs. (46)—(51) can be used in Eq. (5) to give the average Hamiltonian in the laboratory frame in terms of the order parameter tensor. The NMR spectrum arising from Eq. (5) can provide a means to determine the orientational ordering of molecules in uniaxial LC. 

We assume the system under consideration to be a single domain. Then the orientational state of the system can be specified by the order parameter tensor S defined by Eq. (63), The time evolution of S is governed by the kinetic equation, Eq. (64), along with Eqs. (62) and (65). This kinetic equation tells us that the orientational state in the rodlike polymer system in an external flow field is determined by the term F related to the mean-field potential Vscf and by the term G arising from the external flow field. These two terms control the orientation state in a complex manner as explained below. 

In the absence of external fields the suspension under consideration is macroscopically isotropic (W = const). The applied field h (we denote it in the same way as above but imply the electric field and dipoles as well as the magnetic ones), orienting, statically or dynamically, the particles, thus induces a uniaxial anisotropy, which is conventionally characterized by the orientational order parameter tensor (Piin h)) defined by Eq. (4.358). (We remind the reader that for rigid dipolar particles there is no difference between the unit vectors e and .) As in the case of the internal order parameter S2, [see Eq. (4.81)], one may define the set of quantities (Pi(n h)) for an arbitrary l. Of those, the first statistical moment (Pi) is proportional to the polarization (magnetization) of the medium, and the moments with / > 2, although not having meanings of directly observable quantities, determine those via the chain-linked set [see Eq. (4.369)]. 

Let us define the z direction to be the orientation direction of the director n. Then the order-parameter tensor S is... 

Once is computed by somehow solving Eq. (11-3), the birefringence tensor and the stress tensor can be obtained. The birefringence tensor is proportional to vS, where S is the order parameter tensor,... 

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by... 

If we now generalize the definition of director to be the unit vector parallel to the major axis of the order parameter tensor, we find that at vanishingly low shear rates, where the order parameter tensor is nearly uniaxial, this definition reduces to the usual meaning of the term director. ... 

Using these definitions, components of the molecular order parameter tensor can be determined (for example, Sjj is determined by measuring the angle between the molecular z axis and the bilayer normal). The experimental order parameter can be related to the molecular order parameter using the equation ... 

Therefore x yz represent the principal axes of the order parameter tensor. The difference between the dielectric constants (at optical frequencies) for polarizations along the x and z axes is... 

For now we have only tackled the microscopic point of view of the computer simulations. However, at the end one is interested in the macroscopic behavior of the studied system. To obtain the macroscopic physical observables, components of the order parameter tensor, Q p = l/2 3aaap — Sap), are calculated as an average over a number of MC configurations. The order parameter tensor is determined with respect to a fixed eigenframe and then it is diagonalized to obtain the orientation of the director and the magnitude of order parameters. In addition one is interested in the total energy of the system and its correlation functions. Other physical observables can be calculated from those. 

Because the equilibrium order in heterophase systems is characterized by only one nonzero degree of freedom of the order parameter tensor, the fluctuation modes of all five degrees of freedom are uncoupled. Due to the uniaxial symmetry of the phase the two biaxial modes are degenerate and so are the two director modes. If a nematic layer is bounded by walls characterized by a strong surface interaction and a bulk-like value of the preferred degree of order, the fluctuation modes /3j s are sine waves, and their relaxation rates may be cast into... 

Equations (10.38) and (10.39) give a nonlinear integro-differential equation for W, and its mathematical handling is not easy. A guidance of how to proceed is obtained from the phenomenological theory in nematics. De Gennes showed that the dynamics of nematics is essentially described by the Landau theory of phase transition and proposed a phenomenological nonlinear equation fof the order parameter tensor... 

Let M be the dhection towards whidi the polymers have been oriented by the external field. This direction will coindde with the director if the system is in the nematic phase. Since the system will retain uniaxial symmetry around m during the relaxation process, the order parameter tensor is vnritten as... 

In principle, this tensor might be used as orientational order parameter for a uniaxial phase, however, its dimensionless form would be more preferable. Therefore, we normalize anisotropy Xz. to the maximum possible anisotropy corresponding to the ideal molecular alignment as in a solid crystal at absolute zero temperamre. Then we arrive at the order parameter tensor [16]. ... 

Order parameter tensor can be written using the director comprments (ct=x,y,z). 

As already mentioned, for the fixed direction of the nematic director n the shear modulus is absent because the shear distortion is not coupled to stress due to the material slippage upon a translation. The compressibility modulus B is the same as for the isotropic liquid. New feature in the elastic properties originates from the spatial dependence of the orientational part of the order parameter tensor, i.e. director n(r). It is assumed that the modulus S of the order parameter Qij r) is unchanged. In Fig. 8.4 we can see the difference between the translation and rotation distortion of a nematic. 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

This polarization is related to the quadrupolar nature of any uniaxial phase. Fig. 10.10c. In the conventional nematic phase of symmetry Dooh and order parameter tensor Q, each molecule or a building block may, on average, be represented by a quadmpole, and the phase may be characterized by a tensor of density of the quadrupolar moment... 

Where does such a strong scattering anisotropy originate from It is evident that the optical anisotropy of nematic hquid crystals plays the crucial role. In fact, the scattering is caused by fluctuations of the director n, i.e. the local orientation of the order parameter tensor. The local changes in orientation of n imply local changes in orientation of the optical indicatrix. 

In nematic liquid crystals, a convenient measure of orientational order is the order parameter tensor, which may be defined in terms of the orientation of the constituent molecules. If 1 is a unit vector along the symmetry axis of a molecule, then the order parameter tensor is Q defined as... 

The notion [136], and accumulating experimental evidence [139], that the the BPni-isotropic coexistence line might terminate in a critical point has stimulated recent theoretical work by Lubensky and Stark [148], Letting Q = Qij denote the customary order parameter tensor, they assume a new, pseudoscalar order parameter formed from the chiral term in the free energy... 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

In the isotropic phase not too far from Tc, the molecules are still locally parallel to each other. Clearly, the mean values of all elements of the local order parameter tensor Qotp r ) are zero and this tensor describes local orientational fluctuations in the isotropic phase. The free energy density of the system in the Landau-de Gennes theory [6.28] is given by (neglecting the magnetic field term)... 

We return to the bond fluctuation model on the simple cubic lattice (Sect. 2.2), with a potential for the bond angle (9), and consider [123] the specific choice of parameters N = 20,f = 8.0. For such stiff chains, one is also interested in the global nematic order in the system (in the simulations, L x L x L boxes with L = 90 lattice spacings were used [123]), which is described by the 3x3 order parameter tensor = Kronecker delta) ... 

The first NMR evidence of large fluctuations in the magnitude of the nematic order parameter tensor in the isotropic phase above has been obtained by the line-width measurements at 3 MHz in PA A [90]. As Tc is approached from above the line-... 

The tilt of the hydrocarbon chains, common in crystal structures, occurred with a wide distribution in the simulations, and appeared to be correlated with the orientation of the molecular plane (i.e., the C - -C -C + plane)." The correlation is such that the molecular plane preferentially coincides with the plane of tilt (containing the membrane normal and the C -C +i direction). This has the interesting consequence that the order parameter tensor (see Section 3.1.2 for a full description) is not simply given by one component, but has a nonzero off-diagonal element and three different diagonal components. This was also known experimentally by using fluor labels that enabled the resolution of all diagonal elements of the order parameter tensor. Thus the simulations... 

Note that on the microscopic level, nematics are characterized by the fact that the equilibrium orientation distribution function /(u) for the molecules in the directions u is not isotropic. The anisotropy is represented by the order parameter tensor S defined as... 

The set of Eqs. (9.23)-(9.26) provides us with an explicit form of the constitutive equation. Once the velocity gradient tensor L(t) is given, Eq. (9.23) enables us to calculate the order parameter tensor S(t) and then the stresses can be calculated from Eq. (9.26). 

The essence of the Larson-Doi mesoscopic domain theory (Larson and Doi 1991) is summarized below. By multiplying n to Eqs. (9.60) and (9.61) and then taking the average over a mesoscopic length scale, they obtained the following expression for the evolution of the spatially averaged texture order parameter tensor S ... 

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