Bond orientation correlation functions

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...

Above Tj, the sixfold bond orientational correlation function decays exponentially. 

Finally, in analogy to the pair correlation function, we can define a two-point sixfold bond orientational correlation function g5 (r,r2). 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

The pair correlation function g(r) and the bond orientational correlation function g (r) for DRPs are shown in Figs. 60 and 61, respectively. These... 

Figure 61. Sixfold bond orientational correlation function for dense random packings of hard disks.

Bond orientational correlation functions and the associated correlation times for polyethylene have been evaluated by Bahar et al. [10]. The former are determined from the definitions given by Eqs. (8) and (9) where nii in Eq. (9) is replaced by the bond vector Ij. Cross-correlation times for orientations of bonds i and k are obtained as the area under the corresponding OACF decay curves according to Eq. (3). The analysis has also been extended to the domain of normal modes by adopting the transformation... 

The extensions of CONROT that were developed in the context of this work are compared in Fig. 9. The comparison is based on the relaxation of the bond orientation correlation function only, since none of these methods affects global properties directly. These runs were conducted with mixtures of 20% CCB moves and 80% of Concerted-Rotation based nK>ves. Filled drcles are the standard biased CONROT results. Empty squares are the results obtained by... 

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)... 

An early expression derived by Valeur, Jarry, Geny and Monnerie (VJGM) for the second orientational correlation function of bonds is [45, 46, 96]... 

In the part devoted to neutral polymers, we mentioned that semiflexible and stiff chains do not obey the behavior predicted by the Kuhn model. Restricted flexibility of the chain can be caused by the presence of stiff units with multiple bonds or bulky pendant groups, but it can be a result of external conditions or stimuli. In the preceding part, it was explained in detail that repulsive interactions together with entropic forces increase the stiffness of PE chains. Hence, a sudden pH change can be used as a stimulus affecting the stiffness of annealed PE chains. The properties of semiflexible polymers are usually treated at the level of the wormlike chain (WLC) model developed by Kratky and Porod [31]. The persistence length, /p, is an important parameter strongly related to the WLC model and has been used as the most common characteristic of chain flexibility—in both theoretical and experimental studies. It is used to describe orientational correlations between successive bond vectors in a polymer chain in terms of the normalized orientation correlation function, C(s) = (r,.r,+j). For the WRC model, this function decays exponentially ... 

The molecular bond direction is defined by an imaginary line between one pair of the nearest neighbour molecules [23]. This means that the measurement of bond-orientation correlation requires a four-point correlation function. To date no appropriate experimental tool allows us to conduct this kind of measurement. Thus the degree of bond-orientation correlation can only be inferred indirectly. Moreover, no available physical field exists which can couple directly to the hexatic order in the hexatic B phase. Consequently, the heat-capacity measurement is one of the most powerful experimental probes used to investigate the nature of the SmA-SmBhex transition and to complement structural identification by X-ray or electron diffraction. 

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