Chain scattering function

Light scattering teclmiques play an important role in polymer characterization. In very dilute solution, where tire polymer chains are isolated from one anotlier, tire inverse of tire scattering function S (q) can be expressed in tire limit of vanishing scattering vector > 0 as 1121... 

Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

Experimentally, these functions are usually determined only indirectly via the scattering functions of the whole system or the scattering functions of marked chains (see, e.g., [34]). This is one of the advantages of computer simulations over to experiments. However, in order to make significant statements for experimental systems it is always very important to directly compare computer simulations with experimental investigations as well as analytic theories. 

The bond fluctuation model not only provides a good description of the diffusion of polymer chains as a whole, but also the internal dynamics of chains on length scales in between the coil size and the length of effective bonds. This is seen from an analysis of the normalized intermediate coherent scattering function S(q,t)/S(q,0) of single chains ... 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

Anisotropy in polymer solutions is rather rare, but, where it occurs, the effect on the derived molecular weight can be large enough to warrant appropriate corrections. The procedures have been developed by Utiyama124) and by Utiyama and Kurata125). Without a polariser or analyser the normal reciprocal scattering function Kc/Rtf can be measured. It is denoted by Z(0) the form of which contains an anisotropy parameter 5 which is a function of the number of optically anisotropic elements and the principal polarisabilities in the chain ... 

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Fig. 4.28 a Form factor associated to the ds-unit of PB, which is schematically represented in the inset, b and c show the Q-dependence of the amplitude of the relative quasi-elastic contribution of the j -process to the coherent scattering function obtained for rotations of the ds-unit around an axis through the centre of mass of the unit and through the main chain, respectively, for different angles 30° (empty diamond), 60° (filled diamond), 90° (empty triangle) and 120° (filled triangle). The static structure factor S(Q) at 160 K [123] is shown for comparison (dashed-dotted line) (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)... 

The experiments on alkali iodides, PEOx-Nal or PEOx-Lil [316-318] were performed on PEO chains of 23 or 182 (-CH2-CH2-O-) monomers and Orion ratios between 15 and 50. The incoherent scattering from protonated polymers was measured using INI 1C, which yields the intermediate scattering function of the self-correlation. The experiments were performed in the homogeneous liquid phase where the added salt is completely dissolved and no crystalline aggregates coexist with the solution, i.e. at temperatures around 70 °C. 

The results were compared to MD-simulations [317]. Whereas the scattering function of pure PEO could be well described, the dynamics of the salt-loaded samples deviates from the predictions obtained with various electrostatic interaction models. The best but still not perfect and - at least for longer times -unphysical model assumes Hookean springs between chains to simulate the Na-ion mediated transient cross-links [317]. 

Fig. 6.24 Comparison of the scattering from a semidilute PDMS solution under normal polymer contrast (I) revealing the correlation length with the single chain scattering (JJ as obtained by a zero average contrast preparation (see text). The line through 4 represents a Debye function with R =7 nm whereas the line through I corresponds to a Lorentzian (Orn-stein-Zernike) with a correlation length f=l nm. (Reprinted with permission from [325]. Copyright 1991 EDP Sciences)...

Angular scattering functions, PIijI, are computed for subchains located in the middle and at the end of a PE chain. The RIS model developed by Flory et at, (S 004 - S 006) is used for the unperturbed chain. Chain expansion is introduced using a matrix treatment which satisfactorily reproduces several configuration-dependent properties of macromolecules perturbed by long-range interactions. 

The Monte-Carlo method is utilized to investigate the conformational distribution in the central section of a PIB decamer at various temperatures. It is checked that a six-state RIS model based on the two matrices P and Pj constitutes a description of the conformational distribution in PIB. The Monte-Carlo results are in excellent agreement with the experimental data on the average dimensions of PIB chains, as well as with the molecular scattering functions of this polymer in solution and in bulk. 

For Gaussian chains, the expression for the scattering function can be simplified further since then... 

The generating function for enumerating paths of length n is actually the scattering function of isolated primary chains of DP = y. 

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