Orientational correlations

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

Other orientational correlation coefficients can be calculated in the same way as tf correlation coefficients that we have discussed already. Thus, the reorientational coiTelatio coefficient of a single rigid molecule indicates the degree to which the orientation of molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelatio function is the rotational equivalent of the velocity correlation function ... 

Viovy,J.L. and Monnerie, L. Fluorescence Anisotropy Technique Using Synchroton Radiation as a Powerful Means for Studying the Orientation Correlation Functions of Polymer Chains. Vol. 67, pp. 99—122. 

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 ... 

For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion... 

The behaviour of orientational correlation functions near t = 0 carries information on both free rotation and interparticle interaction during collisions. In the impact approximation this information is lost. As far as collisions are considered as instantaneous, impact Eq. (2.48) holds, and all derivatives of exponential Kj(t) have a break at t = 0. However,... 

Fig. 2.5. Time evolution of the second derivative of the orientational correlation function. The long-time asymptotics described by impact theory is shadowed.

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Euler angles Q define the e orientation, and an orientational correlation function of 1th order is introduced in the usual way ... 

It must be stressed that every Un(t,J) brings dq(0) to a different coordinate system. Consequently, the averaged operator (A7.13) is actually a weighted sum of the quantities in differently oriented reference systems. It can nevertheless be used to find the scalar product (A7.7), that is the orientational correlation function. 

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... 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

At the simplest level the orientational correlation of molecular pairs can be characterised by the averages of the even Legendre polynomials Pl(cos J ij) where is the angle between the symmetry axes of molecules i and j separated by a distance r. This correlation coefficient is denoted by... 

It is possible, therefore, to use the GL(r), normally for L = 2 and possibly 4, to explore the extent of the local structure determined by the direct orientational correlations. An alternative and more detailed way in which to investigate the... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

Fig. 8. The distance dependence of the bond orientational correlation function gs (r ) found for the mesogen GB(4.4, 20.0, 1, 1) in the smectic A (.) and the smectic B (—) phases...

Reorientational relaxation times, tJ can be estimated from the assumed exponential decay of the orientational correlation functions cf(/), defined as the average of the / I.egendre polynomial of cos 0, ... 

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation function... 

The method is likely to be useful for the numerical calculation of other correlation functions of importance to complex molecules. An example is the orientation correlation functions of interest in NMR-derived dynamical estimates for proteins and nucleic acids [134], Such correlations are difficult to converge numerically when multiple conformations separated by large free energy barriers contribute to their measurement. 

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... 

Figure 9. Orientational correlation function of the principal axis as a function of center-of-mass distance for freely jointed hard chains for N = 20.

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

Polymer Liquid and Glass. II. Short Range Order and Orientation Correlation. 

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