Bond correlation function

Slow relaxation of water was further discussed in more broad contexts. In particular, since the IS picture is expected to become appropriate as the temperature is decreased, special attention has been paid to water in supercooled states, and dynamics in supercooled states has been investigated in relation to applicability of the mode-coupling theory [51]. It was found that the bond lifetime of individual molecules obeys the thermal process, whereas the bond correlation function shows power-law behavior [52,53], The behavior below or above the temperature at which the mode-coupling theory can be applied was also studied and the transition between IS structures, which is just the network rearrangement dynamics just mentioned above, has clearly been identified in supercooled regions. 

Experimentally, one can use two complementary ways to evaluate the persistence length from the images of the chains. The bond-correlation function... 

FIGU RE 16.6 Temperature dependence of protein and water dynamical properties from MD simulations of a hydrated powder of MBP [8]. (a) MSFs of protein nonexchangeable H atoms averaged over 1 ns blocks of the trajectories, (b) Temperature dependence of the inverse of the correlation times, of the protein-water hydrogen bond correlation functions [73]. (c) Value of... 

We examine this aspect of the spacer by computing the bond correlation function (BCF),

<(3cos 0j -l)/2>. This conforma-tionally averaged function of the angle between the first and jth bond of the chain is an indicator of the persistence of orientational correlations down the chain. For the all- trans conformation of an alkyl chain, for example, the BCF will be unity for j =... 

Fig. 6 The conformationally averaged bond correlation functions of three types of chains the ordinate is shifted for each case.

The bond correlation functions of Eqs. (18) and (19) are not appropriate when Reptation is used, because this method continuously changes the identity of the end bonds. As a result, fp bcr (n) decays rapidly for small n however, this is a spurious decay, associated with renumbering-related decorrelation and the very rapid random changes at the chain ends, while the overall shape of the chain does not change much. The 6nal long-n slopes of the bond correlation functions can still be useful [81]. 

This is similar to the autocorrelation function of the end-to-end distance [82-84], and decays to zero much more slowly than the bond correlation functions of Eqs. (18) and (19) [82-84]. 

Effect of the maximum change in CX)NROT driver angle change First and second degree bond correlation functions plotted vs the number of Monte Carlo steps. A melt of ten C7, polybead chains was simulated with orithms composed of 20% CCB moves (Ncur = 15) and 80% CONROT moves. (O) = W. 

Figure 10 presents bond correlation functions, and end-to-end relaxation functions obtained from longer simulations. Both runs were made with 20% CCB end moves (Ncut = 15), 20% reptation moves, and 60% biased ECROTl moves with Atj) equal to 10°. The first run was started from a system configuration where the constituent chains have conformations close to the unperturbed ones the second run departed from a set of relatively collapsed chain conformations. It is demonstrated that, even with the present algorithm, it is still not possible to obtain fully-equilibrated melts for these long-chain systems. From the different relaxations of bond correlation functions and end-to-end distance vectors (Fig. 10 (top)), it is clear that the results obtained depend strongly on the initial conditions chosen. An examination of the fluctu-... 

Fig. IX Comparaon of polyethylene and polybead 71 chain systems. The id-to-end distance relaxation function and the first and second dt ee bond correlation functions are plotted for (V) polybead, and ( ) polyethylene with explicit hydrogen atoms. Both algorithms were composed of 20% CCB (Ncirr = 15 and 80% CONROT (A ]) " = 10°)...

This form of the bond-bond correlation function indicates that above a threshold value of the Debye screening length, there are... 

In rewriting eqn [90], we have introduced q=for/Nb and substituted summation by integration. In the case of the neutral chain, y(q) = 0, the integral on the rhs of eqn [90] is easily evaluated, resulting in p= 1/K. Substituting average values for the mode amplitudes in eqn [87], one obtains the following equation for the bond-bond correlation function ... 

In the case of the weak electrostatic interactions, we can consider electrostatic potential V(q) as perturbation and expand eqns [90] and [91] in the power series of function V q) (see Reference 144 for details). The analysis of this expansion demonstrates that at the distances along the polymer backbone smaller than the Debye saeening length, Ib/rc < 1, the electrostatic corrertion to the bond-bond correlation function shows quadratic dependence on the distance between monomers I such that... 

A detailed analysis of the fluctuation spectrum in eqn [91] shows that the bond-bond correlation function indeed can be approximated by eqn [79]. At long length scales, the spectrum of fluctuations of the semiflexible polyelectrolyte chain is that of a semiflexible chain with the effective chain s bending rigidity Xi equal to the sum of the bare chain s bending constant and the OSF contribution... 

The amplitude B determining the magnitude of the second exponential term in the bond-bond correlation function scales with the system parameters as ... 

Remarkably, however, this correction alters the dominant behavior of the bond-bond correlation fimction, C(s), as a function of the curvilinear distance, s, along the chain. The conneaion between the mean-squared distance between two segments, n and m, along a polymer and the bond-bond correlation function is given by... 

Like the chain extension, the amplitude of these long-ran d correlations along the polymer chain increases with decreasing film thickness, D, in ultra-thin films. To leading order, the bond-bond correlation function, C(s), is independent of the chain length, N however, the significance on the scale. Re, is suppressed by an additional factor 1/VW in the bulk as compared to ultra-thin films. 

The quality of the quasi-instantaneous field approximation and the ability of SCM F simulations to capture nontrivial correlation effects is illustrated by studying deviations from the Gaussian chain statistics due to the correlation hole effect in polymer solutions [94] and melt [95,96]. These intermolecular correlations result in a power-law decay of the intramolecular, bond-bond correlation function [95, 96] in a one-component melt as shown in Figure 5.4. 

Given the simplicity, it is rather surprising that various dynamic and structural properties of DNA and RNA can successfully be described by this model. One essential feature is that the exponential decay of the bond-bond correlation function defines a characteristic length scale, the persistence length f, as we have already mentioned it,... 

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