Order parameter tensorial

In the spirit of our model, two order parameters play a role the nematic tensorial order parameter Q,j and the smectic A complex order parameter practical reason we use the director fl and the modulus sW in the uniaxial nematic case... 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

Due to external fields the cylindrical symmetry of the order can be lost and the tensorial order parameter becomes somewhat more complicated. 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

Following the lines proposed above will give a prediction of the pattern formed above onset. For a transition from undulating lamellae to reorientated lamellae or to multilamellar vesicles, defects have to be created for topological reasons. Since the order parameter varies spatially in the vicinity of the defect core, a description of such a process must include the full (tensorial) nematic order parameter as macroscopic dynamic variables. 

Although we expect for dimensional reasons that the average magnitude of h, and hence the magnitude of ([nh]), will be proportional to pv, the tensorial form of ([nh]) is unknown. To obtain ([nh]), without having to revert back to (an almost impossible) microscopic calculation of the director field, Larson and Doi (1991 Kawaguchi 1996) assumed Aat ([nh]) is a function of the mesoscopic order parameter S— that is, that ([nh]) = Ka f(S). Dimensional reasoning then leads to the ansatz that... 

Apart from this n-vector model allowing for a -component order parameter, there is also the need to consider order parameters of tensorial character. This happens, for example, when we consider the adsorption of molecules such as N2 on grafoil. For describing the orientational ordering of these dumbbell-shaped molecules, the relevant molecular degree of freedom which matters is their electric quadrupole moment tensor,... 

Another output of Landau theory is that any other physical quantities (tensors) that are coupled to the primary order parameter p contain components that may exhibit also some anomaly. Typically, if has the same symmetry as p, then oc p. Otherwise, ocpm with exponent m = 2 although other values are possible. As a consequence, the phase transitions can be detected in an indirect manner by the measurement of any physical tensorial quantity that is coupled to the order parameter depending on the symmetry of the coupling, some components may become nonzero in the low-symmetry phase, or otherwise exhibit an anomalous behavior near the transition. Schematic evolution of different physical parameters at second-and first-order phase transitions is summarized in Fig. 3. These considerations are highly relevant to NMR because all interactions are second-rank tensors that may couple with the order parameter. 

A tensorial NMR parameter suitable for stress and strain imaging is the strain- and orientation-dependent quadrupole splitting of deuterons. Deuter-ated butadiene oligomers have been incorporated into household rub-berbands by swelling in order to apply spectroscopic imaging and doublequantum imaging of deuterons to detect stress and strain distributions under applied strain [82, 83]. 

We have performed our simulation within the LL lattice spin model, on lattices of 30 x 30 x /i spins, where h — h + 2 represents h layers of nematic LC spins and two additional layers of fixed spins [16]. At the four lateral faces of the simulation sample we have employed periodic boundary conditions to mimic the bulk-like conditions. The standai d Metropolis procedure [56] has been used to update the lattice. The state of a system was monitored by the tensorial nematic order parameter calculated with respect to the fixed frame spanned by the orthonormal triad (e, e, Cz) Q = (3uj Ui - l)/2). 

The order parameter thus characterizes the transition, and the Landau free energy expansion in this order parameter, and in eventual secondary order parameters coupled to the first, has to be invariant under the symmetry operations of the disordered phase, at the same time as the order parameter itself should describe the order in the condensed phase as closely as possible. In addition to having a magnitude (zero for T> Tc, nonzero for T< T ), it should have the same symmetry as that phase. Further requirements of a good order parameter are that it should correctly predict the order of the transition, and that it should be as simple as possible. As an example, the tensori-al property of the nematic order parameter... 

Using second-quantization, it is often necessary to transform complicated tensorial products of creation and annihilation operators. If, to this end, conventional anticommutation relations (14.19) are used, then one proceeds as follows write the irreducible tensorial products in explicit form in terms of the sum over the projection parameters of conventional products of creation and annihilation operators, then place these operators in the required order, and finally sum the resultant expression again over the projection parameters. On the other hand, the use of (14.21) enables the irreducible tensorial products of second-quantization operators to be transformed directly. 

In order to simplify the exposition, we consider that the process is isothermic, and we assume that the evolution of the internal structure can be described with the aid of a scalar-hardening parameter (the density of dislocations or the equivalent plastic strain) and of an internal tensorial parameter (back stress). These restrictions are eliminated in the authors... 

It is possible to identify all of these different contributions to the intensity parameters when deriving their tensorial form explicitly at the second, third, or even higher orders of perturbation approach. It is impossible however to go backwards and select parts of fitted parameters and assign to them a particular physical meaning or interpretation. In general, the parameters Tlx represent the overall picture of possibly important mechanisms that affect the/ <—> f transition amplitudes, without any specification about the order of perturbation, since this particular formalism does not play any role at the point of numerical analysis of their fitted values. 

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