Density correlation functions

Several structure sizes caused by microphase separation occurring in the induction period as well as by crystallization were determined as a function of annealing time in order to determine how crystallization proceeds [9,18]. The characteristic wavelength A = 27r/Qm was obtained from the peak positions Qm of SAXS while the average size of the dense domains, probably having a liquid crystalline nematic structure as will be explained later, was estimated as follows. The dense domain size >i for the early stage of SD was calculated from the spatial density correlation function y(r) defined by Debye and Buche[50]... 

Fig. 4.36 Scaling representation of NSE data (density correlation function) corresponding to PI at Q=1.92 A [second maximum of S(Q)]. Times have been divided by the KWW time Tpair to obtain a master curve. T=230 (cross), 240 (empty circle), 250 (plus), 264 (empty square), 280 (empty triangle), 320 K (empty diamond). The solid line indicates the fit with the KWW law for 250 K

Typically, scaling approaches are employed to explain the behavior in the semidilute regime. By examining static correlations near the temperature, Daoud and Jannick( ) have expressed the density-density correlation function in terms of a correlation length that is inversely proportional to concentration. Since the diffusion coefficient is inversely proportional to the correlation length it is directly proportional to the concentration. 

In the experimental geometry used in this study, the distribution of scattered intensity measured by the detector is the two dimensional Fourier transform of the cross section of the electron density correlation function with a plane perpendicular to the extrusion direction (11). 

The osmotic modulus, K, the frictional coefficient, f, and the diffusion coefficient, D, are related to density-density correlation function of the network, g(r), by [62]... 

Short range order in liquid-like systems as well as long range order in crystalline domains are reflected in WAXS-patterns very dearly. Some examples of calculated X-ray patterns from PTFE (Phase I), a smectic LC-phase and even a PE melt, show that our model covers a wide range of macromolecular structures running the whole scale from crystalline systems over mesophases up to polymer melts. The range of intra- and intermolecular order can be estimated fairly well with the help of density correlation functions. 

Accounting for molecular conformations or torsional and rotational chain dynamics, it is more useful to calculate the mean intramolecular structure factor in terms of density correlation functions (DCF). The structure factor results simply from a Fourier transform of the corresponding DCF. 

The double sum in Eq. (IS) can now be expressed by a cylindrical density correlation function n(r) which counts the number of molecular sites lying in a circular shell of thickness dr with distance r to a given point of reference on the symmetry axis of the monodomam. This RDCF contains the lateral distance statistics of segments within a domain - procedure (3), as well as the average of the domain ensemble - procedure (4). Due to the cylindrical symmetry we finally obtain... 

One of the few attempts to tackle the problem of ionic criticality more quantitatively was made by Hafskjold and Stell in 1982 [36], and was later taken up by H0ye and Stell [17, 302, 303]. Based on a comparative analysis of the correlation functions for nonionic and ionic fluids, these authors asserted that the critical point of the RPM is Ising-like. To this end, they argued that the density-density correlation function hpp(r) and the associated direct correlation function cpp(r) obey essentially the same OZ equation and closure as that of a single-component, nonionic fluid. It was assumed that this analogy suffices to ensure that the critical exponents are Ising-like. 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

When Eq. (230) is replaced in Eq. (228), the density correlation function is found to have two simple poles,... 

Correlations among specific segments can be measured only in computer experiments. Normal scattering experiments measure the segment density correlation function, to which all segments contribute equally. To define this quantity we introduce the local segment, density... 

In later chapters we will see a few more examples of explicit calculations. Specifically we will present the calculation of the single-chain density correlation function in Appendix A 15.2 and the calculation of the osmotic pressure in Appendix A 17,1. 

A more subtle problem occurs for quantities involving several characteristic length scales, Consider for instance the density correlation function in the limit of large momenta (qi ff)2 > 1 where 1/q defines a length scale of interest, which is much smaller than Rg. In the excluded volume limit simple scaling considerations (cf. Sect. 9.1, Eq. (9.20)) suggest... 

Finally let us introduce the density correlation function (second moment) ... 

The (full) density-density correlation function is defined as... 

Drag Resistance in Terms of the Density-Density Correlation Function... 

Here Ai is the imaginary part of the retarded density-density correlation function Ai(k,iv) = Ai —k,u) = —Ai(k, —iv). Equation (7) was derived by different means in [8] similar expressions have been also obtained for noninteracting systems with disorder [9]. Here we demonstrated the validity of Eq. (7) for clean interacting systems. 

The parameter A here is related to the conventional interaction parameter g of the Luttinger liquid g = 1/A. This relation follows from the definition g = vp/u in terms of the velocity of the collective mode (plasmon) u, and its value u = (7m/m)A in the Calogero-Sutherland model [14, 13]. For the rational values of A and at T = 0 the density-density correlation function is known exactly [14, 13]. Due to the integrability of the model, Ai k,w) 0 only in a finite interval of u> around u> = Uik [15]. We found this interval for k < 27rnj ... 

Detailed high-frequency (terahertz) dynamical studies of glasses have been probed by inelastic X-ray scattering (IXS) [139], The advantage of this technique is that with reliable measurements it allows determination of the so-called nonergodicity parameter f(q, T) as a function of wavevector q this quantity is defined by the long time limit of the density-density correlation function F(q, t) divided by the static structure factor [15],... 

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