Melt structure factor

We show typical examples for the melt structure factor and for the single-chain structure factor in Figure 7. The upper panel is for a chemically realistic simulation of PB,111 where the scattering was calculated with the... 

Figure 7 Comparison of melt structure factor and single-chain structure factor for PB (upper panel, calculated as scattering from the united atoms only) and a bead-spring melt (lower panel, in Lennard-Jones units).

We can see from Figure 7 that for momentum transfers larger than about 3 A-1 in PB, i.e., starting around the second maximum, one observes only intramolecular correlations in the melt structure factor112-114 when one considers only scattering from the united atom centers. The melt structure factor can always be decomposed into a chain contribution (Sck(q)) and a contribution that captures the correlations between distinct melt chains (S,nj(c])). 

The chemically realistic simulations we are discussing have been performed using a united atom representation of PB, which leads to the question How does one actually measure a CH vector reorientation for such a model The answer to this question is to use the trick we discussed in the analysis of the pressure dependence of the melt structure factor of PB. Hydrogen atoms are placed on the backbone carbons at their mechanical equilibrium positions for each structure that has been sampled along the MD trajectory. The CH vector dynamics we are showing in Figure 16 is solely from the backbone reorientations of the chain. 

Properties of a Simulated Supercooled Polymer Melt Structure Factors, Monomer Distributions Relative to the Center of Mass, and Triple Correlation Functions. 

Figure 12. Melt structure factor for PB as obtained from simulations of the CRC model and the FRC model at 273 K.

The technique can also be applied to the examination of semi-crystalline polymer systems, for example. Fig. 2.29 shows the WAXS and broad Q neutron diffraction from the same sample of d-PVC. In the X-ray scattering data (Fig. 2.29a) the material appears very disordered, whilst the neutron scattering from the same perdeuterated sample of PVC shows a typical melt structure factor similar to polyethylene although the first peak is somewhat sharper reflecting a higher level of order which perhaps is a consequence of the more polar structure (Mitchell 2011). 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Bowron et al. [11] have performed neutron diffraction experiments on 1,3-dimethylimidazolium chloride ([MMIM]C1) in order to model the imidazolium room-temperature ionic liquids. The total structure factors, E(Q), for five 1,3-dimethylimidazolium chloride melts - fully probated, fully deuterated, a 1 1 fully deuterated/fully probated mixture, ring deuterated only, and side chain deuterated only - were measured. Figure 4.1-4 shows the probability distribution of chloride around a central imidazolium cation as determined by modeling of the neutron data. 

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)...

Fig. 12a, b. Dynamic structure factor for two polyethylene melts of different molecular mass a Mw = 2 x 103 g/mol b Mw = 4.8 x 103 g/mol. The momentum transfers Q are 0.037, 0.055, 0.077, 0.115 and 0.155 A-1 from top to bottom. The solid lines show the result of mode analysis (see text). (Reprinted with permission from [36]. Copyright 1994 American Chemical Society, Washington)... 

Fig. 22. Scaling representation of the spectra obtained from a Polyisoprene melts in terms of the Rouse variable. The solid line displays the Rouse structure factor using the Rouse rate determined at short times, (o Q = 0.153 A"1 V Q = 0.128 A-1 Q = 0.102 A"1. Q = 0.077 A"1 A Q = 0.064 A-1) (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Chemical structure factors affect the melting point and glass transition temperature in much the same manner. A good empirical rale for many polymers is (142-144)... 

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... 

Before we examine in more detail the dynamics of a super-cooled melt of coarse-grained chains and of PB chains, respectively, let us first compare the structure of these two glass-forming systems. Structure is obtained experimentally from either the neutron or the X-ray structure factors. The melt (or liquid) structure factor is given as110... 

The structure factors are Fourier transforms of radial pair-distribution functions for the complete melt or the single chain, respectively,... 

Simulation Study of the a-Relaxation in a 1,4-Polybutadiene Melt as Probed by the Coherent Dynamic Structure Factor. 

Pressure Dependence of the Structure Factor of 1,4-Polybutadiene Melts. A Molecular Dynamics Simulation Study. 

Fig. 3.5 Single chain structure factor from a PEE melt at 473 K. The numbers along the curves represent the experimental Q-values in [A ]. The solid lines are a joint fit with the Rouse model (Eq. 3.19). (Reprinted with permission from [44]. Copyright 1999 American Institute of Physics)...

Fig. 3.6 Single chain structure factor from PEE melts as a function of the Rouse scaling variable. The dashed line displays the Rouse prediction for infinite chains, the solid lines incorporate the effect of translational diffusion. The different symbols relate to the spectra displayed in Fig. 3.5. (Reprinted with permission from [40]. Copyright 2003 Springer, Berlin)...

Fig. 3.9 Dynamic structure factor for a 100 monomer PE chain in the melt at 509 K vs. scaled time for the experiment (symbols), the united atom model (full curves) and the explicit atom model (dashed curves). (Reprinted with permission from [52]. Copyright 1998 American Institute of Physics)...

Fig. 3.11 Dynamic structure factor for a PB melt at 353 K (M =l,600) obtained from simulation (lines) and neutron spin echo measurements (symbols). The Q-values are given adjacent to the respective lines. (Reprinted with permission from [55]. Copyright 2000 Elsevier)...

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