Molecular axis orientational correlation

In order to estimate the orientations of the molecules with respect to the surface, it is convenient to define a molecular axis orientational correlation function, G2(z), by... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Here the summation is over molecules k in the same smectic layer which are neighbours of i and 0 is the angle between the intermolecular vector (q—r ) projected onto the plane normal to the director and a reference axis. The weighting function w(rjk) is introduced to aid in the selection of the nearest neighbours used in the calculation of PsCq). For example w(rjk) might be unity for separations less than say 1.4 times the molecular width and zero for separations greater than 1.8 times the width with some interpolation between these two. The phase structure is then characterised via the bond orientational correlation function... 

In discotic phases the orientation of the molecules is perpendicular to the molecular plane. Here, the columns can be arranged in a nematic or columnar manner. In the nematic phase the molecules possess a centre of gravity randomly ordered, but with the short molecular axis of each molecule more or less parallel. In the columnar phase, beside the preferable orientation of the short molecular axes, the disc-like molecules are ordered forming columns. Depending on the correlation strength between he columns these phases can be subdivided into ordered or disordered. A third possibility is to have a thermodynamically preferable position of the columns in the mesophase, like in a hexagonal cell. Additionally, a tilt of the columns is also possible. 

The functions X(Q) and Y(fl) are specified by the choice of the particular experiment. Prominent orientational correlation functions result when setting X(Q) = K(Q) = P/(cos0), where P/ is the Legendre polynomial of rank / and the angle 0 specifies the orientation of the molecule with respect to some fixed axis. For example, consider a molecule that possesses a vector property, say the molecular electric dipole p = pu, (u is a unit vector). Then, one defines the dipole autocorrelation function g (t) = (u,(t)u,(0)). Similarly, one defines a correlation function gilt) for second rank tensorial molecular properties. In general the normalized (g/(0) = 1) orientational correlation function of rank l is given by... 

Mobility in this region is dominated by short-time motion, typically < 2 ps. After that time, all correlation of molecular motion is lost due to frequent collisions with the cavity walls. The center-of-mass velocity autocorrelation function of the penetrant exhibits typical liquid-like behavior with a negative region due to velocity reversal when the penetrant hits the cavity wall [59]. This picture has recently been confirmed by Pant and Boyd [62] who monitored reversals in the penetrant s travelling direction when it hits the cavity walls. The details of the velocity autocorrelation function are not very sensitive to the force-field parameters used. On the other hand, the orientational correlation function of diatomic penetrants showed residuals of a gas-like behavior. Reorientation of the molecular axis does not have the signature of rotational diffusion, but rather shows some amount of free rotation with rotational correlation times of the order of a few tenths of a picosecond, although dependent in value on the Lennard-Jones radii of the penetrant s atoms. 

In this expression T20 is determined by the way in which the shape of one molecule distorts the radial distribution function away from spherical symmetry whilst T22 is determined by mutual orientational correlation between molecules. The distribution function of model diatomics obtained in computer simulations showed that the T22 effects were much smaller than the orientation-position effects given by T2q The effect of the T2q terms on Yeff sn be simply interpreted for a cylindrical molecule the closest neighbours must lie around its waist. If the cylinder is aligned with an external field (ie if the field is parallel to the molecular z axis) then the dipole induced by the field in the near neighbours will tend to reduce the net field along the cylinder axis as the dipole induced in the cylinder by a given external field is thereby reduced the effect is to reduce Ugz Conversely if the cylinder is perpendicular to the field (ie parallel to x) the dipole... 

Fig. 9 A. Schematic representation of orientational correlation functions leff a medium without rotational freedom, such as a crystal or a glass centre, a system with full rotational freedom, a proper liquid righf a hquid crystal in which correlation is preserved only along one molecular axis.

Before we leave ellipsoids, we note that it is also possible to stretch or compress along a second axis, to leave a biaxial ellipsoid. Allen has studied systems of biaxial ellipsoids, in which the molecular axes are all of differing lengths (a b c) [18]. At the rod limit (1 1 10) and disc limit (1 10 10), uniaxial nematic phases are observed, as expected. In between these limits (1 < b < 10), the molecule is biaxial (1 b 10) it is, therefore, possible that these model molecules could exhibit a biaxial nematic phase, in which the orientations of all three molecular axes become correlated at long range. The simulations provide evidence that such a system does indeed exhibit isotropic, uniaxial nematic and biaxial liquid crystalline phases. The biaxial phase is found to be most stable when... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

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