Orientation correlational 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... 

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... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

Usually the available Hv data are insufficient for a Fourier inversion of Eq. (IV-34), needed to obtain the orientation correlation function Thus, one is faced with the necessity of reverting to simple models... 

Another experimental method that has been used to determine orientational correlation functions in macromolecular systems is based on measurements of the time-dependence of the depolarization of fluorescence 26 From these measurements rotational diffusion coefficients and the shape of the rotating macromolecule have been determined.27... 

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

Third, one needs accurate static orientational correlation functions. These are now available for the first rank correlation functions for water, acetonitrile, and several other liquids [191]. 

Figure 3 (A) Spectra of CO in a variety of environments and (B) corresponding orientational correlation functions. In (A), the four curves correspond to (top to bottom) CO in the gas phase, in cyclohexane, in water, and in an orientationally constrained environment (theoretical spectrum with a = 0.5 see text). In (B), the corresponding ordering is bottom to top. Note the similarity of the fast, inertial contribution to the orientational correlation function at early times.

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