Collective structure factor

Fig. 5.9. Plot of the static collective structure factor S(q) for BPA-PC at T = 570 K, as obtained from mapping to the bond fluctuation model. From [175]...

The defining property of a structural glass transition is an increase of the structural relaxation time by more than 14 orders in magnitude without the development of any long-range ordered structure.1 Both the static structure and the relaxation behavior of the static structure can be accessed by scattering experiments and they can be calculated from simulations. The collective structure factor of a polymer melt, where one sums over all scattering centers M in the system... 

To monitor the time evolution of the long-range ordering, the collective structure factor at the constant time interval of phase separation is calculated by Eq. (20) ... 

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

At this point we emphasize that Eqs. (16H33) should not be understood as a formally rigorous derivation (such derivations by various techniques can be found in the literature [114, 116, 117]) but rather the present treatment (which follows Ref. 78) is a plausibility argument. In this spirit, one can also extend the theory to the interacting case, within the framework of mean field theory the inverse collective response function of the interacting system within RPA is always found from that of the noninteracting system by subtracting the Fourier transform of the interaction [118]. In our case we have a collective structure factor... 

Eq. (54) is a function of vA/vB only [130]. We also note the corresponding generalization of the collective structure factor,... 

Fig. 18. Schematic evolution of the order parameter probability distribution Pl(M) from T < Tc (bottom) to T = Tc (middle) toT > Tc (top), and corresponding temperature dependence of the order parameter < i = < M >, collective structure factor at zero wave vector kBTic = Ld M2> — < M >2), d being the dimensionality of the box, and the reduced fourth-order cumulant UL = 1 — l/(32). Dash-dotted curves indicate the singular variation which results in the thermodynamic limit, L-> oc. From Binder [258]...

Fig.44. Collective structure factor S(x,e) dotted vs x = qRg(e,N) for f = 1/2, N = 20 and various choices of the energy kBTe between monomers of different kinds, allowing for a volume fraction , = 0.2 of vacancies on the simple cubic lattice. Curves are a fit to Eq. (187), treating both % and Sg in Eqs. (187) — (189) as adjustable parameters, while the actual gyration radius is used for the normalization of the abscissa. Perpendicular straightline shows the value x = 1.945of Leibler s theory [43]. The symbols denote the choices eN = 0,1,2,3,4 and6 (from bottom to top). From Fried and Binder [325],...

The finite size also causes a discreteness of the possible values of q to (2n/L) (vx, vy, vz), where (vX) vy, vz), are integers, in the calculation of the collective structure factors, which in the simulations is normalized as... 

Collective structure factor 194-203, 214, 233, 251, 259, 272, 273, 280 Complexation in polymerization 148 Convective term 226 Crown ethers 9 Cyclohexane/methanol 213... 

Combined with Eq. 26, this yields the well-known RPA expression for the collective structure factor in binary polymer blends... 

The partial density-density collective structure factors are given by... 

Figure 24. Predictions of PRISM theory with the PYclosure for the low wave vector common block collective structure factor for the athermal copolymer models listed in Table II. Note that the common block structure factor intensifies monotonicaily as the overall copolymer stiffness stiffness increases due to increasing B-block aspect ratio.

The linearized R-MPY version of the thread molecular closure condition of Eq. (6.7) can be shown to result in a nonlinear, self-consistent integral equation for the effective chi parameter [or equivalently the concentration fluctuation part of the collective structure factor 5(A )]... 

Figure 39. Total collective structure factor of a dense melt of / x lOO stars. Note the logarithmic scales.

The accuracy of the united atom description for polyethylene has also been carefully tested in the literature [60, 71], both by comparison of simulation results with experiments and with simulations dealing with an all-atom model where hydrogen atoms are explicitly considered. Of course, for polyethylene the vapor liquid critical point would occur at very high temperatures, where the macromolecule would no longer be chemically stable, and is of no physical interest thus one uses data for single chain and collective structure factors to gauge the accuracy of the simulation models in this case. 

In this situation, most Monte Carlo work has emphasized the behavior of the collective structure factor and of the chain linear dimensions 24,52-55,84,85 disordered phase of the bulk copolymer melt. Assigning... 

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