SANS, form factors

Figure 3 Example of SANS curves at two times of the reaction. The lines are calculations of the form factor. (A) prior to TEOS addition, the micelles are well described by core-shell spheres, with an external radius of 7.1 ran. ( ) 15 minutes after the beginning of the reaction, the micelles can be viewed as cylinders of length 50 nm and radius 6.9 nm.

We tried to extend the network model to the binary blends and to determine the number n of elastic strands per S2 chain by adjusting the form factor calculated from the recoverable strain to the SANS data. Fig. 25 gives an example of the type of agreement which could be obtained between the model and the data. The corresponding values of n are listed in Table 6. 

The theoretical description in terms of spherical harmonics also yields a relation between the size polydispersity index p of the microemulsion droplets and the bending elastic constants [43]. The quantity p is accessible by SANS [51, 52, 59-61]. For polydisperse shells as obtained by using deuterated oil and heavy water for the preparation of the microemulsion (contrast variation), one can account for the droplet polydispersity by applying an appropriate form factor, e.g. containing a Gaussian function to model the size distribution [52, 59, 62]. A possible often-used choice is the following form factor... 

The Debye function is the scattering form factor of a single polymer chain in a melt [88, 92]. Experimentally, this function can be observed using the method of contrast variation in SANS. A calculation similar to the one presented above can be done for soft matter objects of arbitrary shape and a lot of form factors were already derived. A good review for different form factors was given by Pedersen [94]. 

Note that in a subsequent study a similar result of micro-/nanophase separation has been observed in block copolymers of PNIPAAM and Ai-isopropylmetAa-crylamide (PNIPMAM). This small-angle neutron scattering (SANS) study used a scattering analysis with a new form factor model taking into account a nanophase separated internal morphology [50]. 

Polydispersity has a profound influence because it smears out the deep minima or zeros of the form factor. For standard deviations above 15% the SAXS-analysis of latex particles becomes very difficult because in these cases the minima of P(q) have nearly disappeared. The same holds true for the SANS-analysis of such systems, of course. 

Fig. 19 (a) Normalized SANS curves in different D20/DMF-d7 compositions at a polymer volume fraction of 0.25%. Solid lines represent fits with a spherical core shell model. Data in pure DMF-dv were fitted with a Beaucage form factor, (b) Aggregation number P plotted versus interfacial tension, y. The solid line depicts the power-law dependence, P rj as predicted by Halperin for star-like micelles [40]. Reprinted with permission from [45]. Copyright (2004) American Chemical Society... 

As the form factor analysis provides detailed insight into the particle structure, one can use this information in order to adapt the polymerization conditions, as, for example, batch, semibatch synthesis, or controlled monomer feed, in order to tailor particle morphology and thus microgel properties.In addition, one can employ SANS data to investigate the influence of chemical reactivity on the microstructure by comparing, for example, different cross-linkers, initiators, or monomers. 

Figure 29 Top SANS data and form factor fits of a PNIPMAM-PNIPAM(50/50) copolymer microgel in D2O at different temperatures two data sets are shifted verticaiiy. Bottom schematic drawing of the internal nanophase-separated dirty snowball morphology of the PNIPMAM-PNIPAM(50/50) copoiymer microgei in D2O at the transition temperature. Note that the fuzziness of the microgel surface is not illustrated.

FIGURE 5 SANS from blend of deuterated and conventional polyisoprene (M = 10 g/mol), plotted in the Gunier form [Eq. (9)] yielding a straight at small angles. The solid line is the fit to the Debye form factor for Gaussian coils [Eq. (lO)j. (From Akcasu, Summerfield, Jahsan et al. [126].)... 

Three parts will appear in the review. First the background of the experiment technical remarks and the possible extensions or alternatives for using SANS in dynamics (Sect. 2), the background of the present interpretation of the data (Sect. 3), and a brief recall of the Doi-Edwards model. Second, a detailed description is given of the different characterizations of the q, t) dependence, radius of gyration, asymptotic laws and comparison of the form factor with some calculated expressions some recent results on crosslinked melts and mixtures of different molecular weights will be presented. The third part contains additional remarks on other more detailed approaches. 

To determine the required time range, it is necessary to first compare the classically obtained times from the viscoelastic behaviour (Fig. 1). The situation is complicated by the fact that these times depend on the temperature. The usual glass time temperature superposition (see Sect. 2) has been checked in the 50s for rheological data the characteristic times T,, etc. depend on the temperature T by the same factor a(t) = 10(Cj(T — T )/Cj -F T — T. For the SANS data, a similar superposition has been proposed a form factor for t = tj at T, is compared to a form factor for (ij )t, at Tj. A satisfying overlapping is possible by a correct choice... 

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