Chain structure factor

WAXS the scattering is completely described by the interactions of neighboring atoms along a single chain, the so-called single-chain structure factor. Cf. descriptions of the Ruland method [14] in textbooks [7,22],... 

Performing neutron scattering not on perdeuterated samples but on a single deuterated chain in a protonated matrix (or vice versa both ways provide the same contrast) gives the single-chain 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).

Neutron Scattering and the Single Chain Structure Factor. . . . . 209... 

Single Chain Structure Factor for Star-Polymer Dynamics.221... 

Neutron Scattering and the Single Chain Structure Factor... 

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)...

Inserting the Rouse rate W(, 3Q9 K)=(7 0.7)xl0 AVns (Table 3.2) obtained from single chain structure factor measurement into Eq. 3.18 the solid line is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also reveal clearly a transition to a law, though Eq. 3.36 would predict the dotted line. The discrepancy explains itself in considering the non-Gaussian character of the curve-linear Rouse motion (Eq. 3.38). Fixing and d to the values obtained from the single chain struc-... 

Fig. 3.25 Comparison of the experimental dynamic single chain structure factors for PEP at Q=0.135 Qd=6A) and PE at Q=0.128 A (Qd=5.5) with the dynamic structure factors from the computer polymer. The various/w// lines represent MD results for different Qd=3.1 (a), 3.9 (b), 4.6 (c), 6.2 (d), and 7.7 (e). In the upper part the computer results are the structure factors from a fully labelled chain, while in the lower part only the centre 35 monomers are labelled. (Reprinted with permission from [49]. Copyright 1992 American Chemical Society)...

The single-chain structure factors calculated in the previous sections correspond to the infinite dilution limit. This limit also corresponds to zero scattering intensity and is not useful so that concentration effects have to be included in the modeling of polymer solutions. First, Zimm s single-contact approximation [5] is reviewed for dilute polymer solutions then, a slight extension of that formula which applies to semidilute solutions, is discussed. 

Effects due to macromolecular shape changes during single-contact interactions can, within the Benoit-Benmouna theory, be included in an ad-hoc fashion by renormalizing the single-chain structure factor to make it concentration dependent. This approach is often used to describe polymer solutions up to the concentrated region. 

The first two terms correspond to the combinatorial entropy terms of Eq. (1) and form the non-interacting part of the structure factor which is just a weighted average of the single-chain structure factors SA(q) and SB(q) of both blend components. SA(q) and SB(q) are characterized by the radius of gyration RgA= aA(NA/6)1/2 and RgB=aB(NB/6)1/2, where aA and aB are the statistical segment lengths of polymer A and polymer B, respectively. The last term of Eq. (4) yields the SANS determined interaction parameter %SANS ... 

Finally, we will briefly discuss the properties of polymer blends under shear flow. In small molecule mixtures, shear flow is known to produce an anisotropy of critical fluctuations and anisotropic spinodal decomposition [244, 245], In polymer mixtures, the shear has the additional effect of orienting and stretching the coils, thus making the single-chain structure factor anisotropic. In the framework of the Rouse model these effects have been incorporated into the RPA description of polymer blends [246, 247]. Assuming a velocity field v = yyex, where x, y, z are cartesian coordinates, y the shear rate, and ex is a unit vector in x direction, the single chain structure factor becomes [246, 247]... 

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