Collective scattering function

We also note that for the collective scattering function S ntO) and correlation length it makes sense to consider both amplitudes f(N), (N) referring to... 

Now the finite size scaling hypothesis [262-264] implies for the collective scattering functions... 

There are still two caveats that must be mentioned, however all measurements refer to an analysis of the collective scattering function ScoM(q) in the one phase region of the blends, thus the precise value of Tc has to be treated as a fitting parameter and does not result from an independent measurement. Secondly, there are problems with understanding the temperature where the crossover from mean field behavior to Ising-like critical behavior occurs, as already discussed in the last section. 

The idealized symmetric blend model is not representative of the behavior of most polymer alloys due to the artificial symmetries invoked. Predictions of spinodal phase boundaries of binary blends of conformationally and interaction potential asymmetric Gaussian thread chains have been worked out by Schwelzer within the R-MMSA and R-MPY/HTA closures and the compressibility route to the thermodynamics. Explicit analytic results can be derived for the species-dependent direct correlation functions > effective chi parameter, small-angle partial collective scattering functions, and spinodal temperature for arbitrary choices of the Yukawa tail potentials. Here we discuss only the spinodal boundary for the simplest Berthelot model of the Umm W t il potentials discussed in Section V. For simplicity, the A and B polymers are taken to have the same degree of polymerization N. 

In Section 7.2.4 it was shown that via a finite size scaling analysis a meaningful extrapolation of simulation data to the thermodynamic limit is possible, and in this way one can extract estimates for both critical exponents (/ , 7) and amplitudes C+(1V), C- N) and B N) of the collective scattering function above Tc (eq. [7.16]) and below Tc -ScoiK = 0) =C- N)r, t = I — T/Tc — 0 or the order parameter (/>" )= B lf)t, respectively. While for short enough chains N < 32), data both for the simple self-avoiding walk model of Fig. 7.3 and for the bond fluctuation model are nicely consistent with the expected critical exponents for the three-dimensional Ising models /3 0.325,7 Pi 1.241), for N>64 one rather finds effective exponents ... 

Finally we mention that also the collective scattering function 5coii(0) and the correlation length f ( crit) must show a crossover scaling... 

Percus-Yevick (PY) 220, 222, 225 R-MMSA/R-MPY 222, 236 Cloud point curve 43 Cluster size, simulation 159 equilibrium 199 Cluster size distribution 155 Cluster, critical 163 CO2 4, 18, 19, 37, 43, 82, 83, 88, 91 Coarse-grained models 94-96 Coarse-graining procedure 22 Cohesive energy density 232, 252 Collective scattering function 13 Collision frequency 124 Colloidal suspensions, crystallization 152 Colloids 54,149, 243 hard-shere 164 weakly charged 176 Compressibility 223, 232, 243 isothermal 229... 

The 3D MoRSE code is closely related to the molecular transform. The molecular transform is a generalized scattering function. It can be used to predict the intensity of the scattered radiation i for a known molecular structure in X-ray and electron diffraction experiments. The general molecular transform is given by Eq. (22), where i(s) is the intensity of the scattered radiation caused by a collection of N atoms located at points r. ... 

Fig. 6.18 Normalized intermediate scattering function from centre-labelled 18-arm PI solutions. Collective corresponds to a 4.85% solution of labelled stars, whereas the self data stem from a solution of 1% labelled and 16.6% non-labelled stars. Note the maximum Fourier time of 350 ns (A=1.9 nm), which was obtained at the INI 5 in the case of these strong scattering samples. (Reprinted with permission from [304]. Copyright 2002 Springer)...

Figure 9. Time dependencies of the single-particle and the collective intermediate scattering functions compared for two different solute sizes at a particular wavenumber q = 6.001 at reduced temperature T" = 0.75 and in the normal density regime (p = 0.89). The solid line represents the collective intermediate scattering function. The long-dashed line is the single-particle intermediate scattering function for solute-solvent size ratio 1.0 and the short-dashed line is for solute-solvent size ratio 0.5. The plots show that the decoupling of the solute motion from the solvent dynamics increases as the solute size is decreased. The time is scaled by rsct where TJC = [mo2/kBT] 2.

A more complete description of the crossover from two- to three-dimensional critical behavior, that incorporates the shift of Tc, Eq. (129), can be written down in terms of crossover scaling functions M for the order parameter M, Eq. (130) and S for the collective scattering intensity Scoll for scattering wavenumber q—>0, which can be accessed by small angle scattering techniques [186]. Defining =1—T/Tc (D->°o) we can write... 

X-ray diffraction by a crystal arises from X-ray scattering by individual atoms in the crystal. The diffraction intensity relies on collective scattering by all the atoms in the crystal. In an atom, the X-ray is scattered by electrons, not nuclei of atom. An electron scatters the incident X-ray beam to all directions in space. The scattering intensity is a function of the angle between the incident beam and scattering direction (26). The X-ray intensity of electron scattering can be calculated by the following equation. 

Fig- 7 Intermediate scattering function C q,t) for a suspension of PMMA hard spheres (radius R = 118 nm) recorded at R = 2.95 and volume fraction (/> = 0.42 in a semi-log and log-log representation (lower inset). The static structure factor is shown in the upper inset where the vertical line indicates the value qR = 2.95 at which the function C(g,/) was recorded. The short-time collective diffusion is obtained from the initial plot of C(, i) (tetl line)... 

The Minsk conferences on chemical bonds in semiconductor and other crystals have demonstrated clearly the importance of experimental determinations of the distribution of the electron density in crystals, the distribution of the potential in the crystal lattice, and the application of various methods to the calculation of the effective charges of ions and of accurate values of the atomic spacings and bond energies. It has been found possible to estimate various physical properties of crystals from the experimentally and theoretically determined atomic scattering functions and the electron density distributions in crystals. These problems are considered in several papers in the present collection. 

This section deals with the fundamental basis of the SCGLE theory. We first describe what is understood here for the GLE and then illustrate its use in the derivation of exact result for the time-dependent friction function A (t), and for the collective and self intermediate scattering functions. In addition, we discuss two additional approximations that convert these exact results into a closed self-consistent system of equations. 

Let us now describe the application of the GEE formalism to the description of collective diffusion. As a result, we shall derive exact memory-function expressions for the intermediate scattering function F(k, f) and for its self-diffusion counterpart F k, t). We then explain the approximations that transform these exact results in an approximate self-consistent system of equations for these properties. 

Furthermore, in order to calculate the total intensity of the collected scattered light, the number of molecules which undergo enhanced scattering must be included in the scattering equation. If the only enhancement mechanism is through the electromagnetic factors, L (o)L (Ojp), a calculation of the scattering intensity would have to take into account the distribution of molecules as a function of distance, r, from the surface of the metal particle, since L r) is a function of distance. This effect would introduce the factor... 

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