Correlation functions dynamic structure factor

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

The self-correlation function leads directly to the mean square displacement of the diffusing segments Ar2n(t) = <(rn(t) — rn(0))2>. Inserting Eq. (20) into the expression for Sinc(Q,t) [Eq. (4b)] the incoherent dynamic structure factor is obtained... 

Fig. 4. Neutron spin echo spectra for the self-(above) and pair-(below) correlation functions obtained from PDMS melts at 100 °C. The data are scaled to the Rouse variable. The symbols refer to the same Q-values in both parts of the figure. The solid lines represent the results of a fit with the respective dynamic structure factors. (Reprinted with permission from [41]. Copyright 1989 The American Physical Society, Maryland)...

Though the functional form of the dynamic structure factor is more complicated than that for the self-correlation function, the data again collapse on a common master curve which is described very well by Eq. (28). Obviously, this structure factor originally calculated by de Gennes, describes the neutron data well (the only parameter fit is W/4 = 3kBT/2/C) [41, 44],... 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

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)... 

In [189] a simple two state model for the dynamic structure factor corresponding to the Johari-Goldstein jS-process was proposed. In this model the jS-relaxation is considered as a hopping process between two adjacent sites. For such a process the self-correlation function is given by a sum of two contributions ... 

Fig. 5.23 Time evolution of the three functions investigated for PIB at 390 K and Q=0.3 A"h pair correlation function (empty circle) single chain dynamic structure factor (empty diamond) and self-motion of the protons (filled triangle). Solid lines show KWW fitting curves. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

Figure 18. (a) Response versus the dynamical structure factor for the binary mixture Lennard-Jones particles system in a quench from the initial temperature Ti = 0.8 to a final temperature T( = 0.25 and two waiting times t = 1024 (square) and = 16384 (circle). Dashed lines have slope l/Tf while thick hues have slope l/T (t ). (From Ref. 182.) (b) Integrated response function as a function of IS correlation, that is the correlation between different IS configurations for the ROM. The dashed fine has slope Tf = 5.0, where Tf is the final quench temperature, whereas the full lines are the prediction from Eq. (205) andF = F (T ) Teff(2") 0.694, Teff(2 ) 0.634, and 7 eff(2 ) 0.608. The dot-dash line is for t , = 2" drawn for comparison. (From Ref. 178.)... 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

Fig. 2.49 Dynamic structure factor at two temperatures for a nearly symmetric PEP-PDMS diblock (V = 1110) determined using dynamic light scattering in the VV geometry at a fixed wavevector q = 2.5 X 10s cm-1 (Anastasiadis et al. 1993a). The inverse Laplace transform of the correlation function for the 90 °C data is shown in the inset. Three dynamic modes (cluster, heterogeneity and internal) are evident with increasing relaxation times.

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

The longitudinal current correlation function, Q(q, t), is related to the dynamic structure factor by the following expression ... 

E (Q, t) is referred to as the intermediate scattering function, and the bars placed above the scattering length variables bj and bk in equation (4) represent the time average of the cross-correlation functions. These scattering lengths are time independent and thus can be separated out from the time variations in the nuclear density. With this in mind, the dynamical structure factor can be derived using the time Fourier transform of E(Q, t) as... 

A recent develtyiment in the theoty ftx- the dynamics structure factor of molecular liquids, which employs the interaction-site modd, is outlined. The theory is applied for a d cription of the solvation dynamics associated with a photo-excitation of a molecule in polar liquid. Preliminary results of the solvation time correlation functions for an atomic molecule in a variety of solvents are presented. 

Concerning the dynamic structure factor, we shall confine our attention to the incoherent case, where the self-correlation function B(0, ) only is required since we have q -4 1, it may be shown that the results are essentially valid for the coherent case as well (long-time limit) [86]. From,Eqn. (3.1.18) we get... 

When the bead-and-spring chain is not in the ideal state, the intramolecular force is given in Eqn. (3.1.3). As it may be seen, in general, the force is not simply transmitted by first-neighboring atoms, but it has a long-range character. The relaxation times are given by Eqn. (3.1.11) after they are known, the dynamic viscosity i (cu) and the atomic correlation function B(k, t) are obtained from Eqs. (3.1.15) and (3.1.18) (for the periodic chain), and the complex modulus and dynamic structure factors are easily constructed. 

The correlation function B(k, t) cannot be evaluated in closed form in the present case. However, for t comprised between T ,j and numerical calculations [54] show that B(0, t) cc whereas, for t x q), B(0, t) oc t due to globule diffusion. Consequently, the exponent of the relationship hiiQ = const has values 3 and 2 in the two respective regimes, as with the unperturbed chain [see Eqs. (3.2.15 ) and (3.2.16)]. We do not obtain any plateau of B 0,t) in this case, unlike the Rouse limit. Figure 7 shows the coherent dynamic structure factor S Q, t) as a function of t for two different Q values both for the unperturbed and for the collapsed chain. The two Q s correspond to observation distances /Q [ref. 15, note 6] just below and... 

MCT is a popular liquid viscosity theory (Gee, 1970 Gotze et al., 1981 Leutheuser, 1984, Jackie, 1989). It is based on the description of the dynamical properties of density fluctuations in terms of a dynamical structure factor. There are inherent density fluctuations in liquids, which decay with characteristic relaxation times. The decay becomes slower as the temperature is lowered due to increase of viscosity. It is controlled by dynamically correlated collisions. The equations governing the decay are non-linear. Analysis of the non-linear equation of motion of the density fluctuations gives a density correlation function of the type... 

The van Hove correlation function and the dynamic structure factor The following section is mainly based on the textbook by Egelstaff [93] and in parts on other standard texts. Again, we use the monoatomic liquid as example. In Section 2.5.2.1, only the spatial correlation of the different positions of the atoms in a liquid were discussed. 

A little work seems to have been carried out on the wavenumber-dependent orientational correlation functions C/m(, t). These correlation functions can provide valuable insight into the details of microscopic dynamics of the system. A molecular level understanding of C/m(, t) would first require the development of a molecular hydrodynamic theory that would have coupling between C/m(, t) and the dynamic structure factor S(k, t) of the liquid. A slowdown in C/m(, t) may drive a slowdown in the dynamic structure factor. This would then give rise to a two-order parameter theory of the type develops by Sjogren in the context of the glass transition and applied to liquid crystals by Li et al. [91]. However, a detailed microscopic derivation of the hydrodynamic equations and their manifestations have not been addressed yet. 

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