Kratky plots

Fig. 47a, b. Structure and dynamics of star-shaped polymers with different functionalities, a Kratky plot of the static structure factor (S(Q, 0) Q2 vs. Q Rg. b Q(Q)/Q3 vs. Q Rg, as derived from Eqs (94) and (123), assuming Rouse dynamics... 

As a signature of the star architecture the elastic scattering data of the completely labelled stars exhibit a pronounced peak at z = 1.5 in the scaled Kratky representation. In contrast to the PI systems, presented before, the Kratky plot of the measured scattering curve disagrees strongly with the prediction of Eq. (123). The experimental halfwidth of the peak is nearly only 50% of the theoretically predicted one. 

Similarly to the fully labelled stars [150], the peak structure in the generalized Kratky plot for core and average contrast conditions is also quite well reproduced by calculating S(Q) on the basis of an ideal (v = 1/2) star conformation. The swollen conformation calculations predict a more diffuse peak in both cases. 

Fig. 55a-c. Generalized Kratky plot S(Q) Q5/3 vs. Q for diblock PIP stars under different contrast conditions. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)... 

At this position, the reduced relaxation rate Q(Q)/Q3 shows a minimum for stars, where the arms are labelled completely or to 50% at the core site. For these systems the Kratky plot of the static structure exhibits a maximum at the same Q-value... 

Figure 11.9 Kratky plots of G1 PS dendrigraft, G3 PS dendrigraft, G4 PAMAM dendrimer and G10 PAMAM dendrimer. Flat high q shows star-like structure, higher order features shows sphere-like structure...

Figure 11.15 Kratky plot of SAXS of G5 PPI dendrimer at various mass fractions. Higher order feature moves to higher q, suggesting shrinkage with increased concentration...

Fig. 3. Form factors. Solid line Linear chain dashed line f=6 star chain dotted line f=12 star chain. Inset Kratky plots (same notation)...

Fig. 4. Generalized Kratky plot of the experimental form factor of an 18-arm PI star (points) solid line fit to the Benoit function, Eq. (23) dashed line fit to the RG curve described in [66]. Reprintedwith permission from [67]. Copyright (1994) American Chemical Society...

The experimental data corresponding to one labeled arm in stars of f=12 (good solvent) [68] shows, as expected, Kratky plot ordinates that increase monotonously with q. However, the plateau is only obtained with an apparent critical exponent of 2/3 (i.e., greater than the theoretical value, v-3/5). This seems an indication of the arm stretching effect, though the scaling and RG theoretical predictions describe this effect only in terms of a pre-exponential factor [11,42]. 

Fig. 16. Generalized Kratky plot of the form factor of a star from numerical data of a MG simulation [161]. x=q . e/kgT=0.1 corresponds to EV chains ande/kgT=0.3 is close to the theta state...

Fig. 7. Normalized scattering intensities I (q)/I (0) of a dendrimer of fifth generation G5 [23] in a Kratky-plot vs q o.- The triangles mark the data measured at highest contrast (in fully deuterated DMA) normalized to unity at q = 0. The squares give the term I q) of...

Finally we mention the Kratky plot which also may help to detect branching. Here ((S2) q2) Pz (q2) is plotted against q. Figures 26 and 27 show the Kratky plots for regular star-molecules and for the soft sphere model. Linear randomly coiled chains result in... 

Fig. 27. Kratky plot for the soft sphere model931. The points represent light-scattering measurements form a PVAc microgel (compare also Figs. 19, 24 and 25)...

Fig. 28. Kratky plot for vulcanisates with different primary chain distributions. Solid curves for a/b = 6 (a = persistence length, b = bond length) and dashed curves for a/b = 1 (Gaussian chains). The numbers in the figure denote the value of Pwp/Pno, i.e. the polydispersity of the primary chains"9) X2 = q2(S2),...

Fig. 29. Zimm-plot and Kratky-plot from a star-branched macromolecule. Star centre nodule of densely cross-linked polydivinyl benzene rays monodisperse polystyrene chains. Measurement in toluene at 20 °C and with the three wavelengths 20 = 546, 436 and 365 nm"1...

Fig. 30. Zimm-plot and Kratky-plot of glycogen from edible mussels. Neutron small-angle measurements in deuterated water. The dotted line describes the scattering behavior of the ABC model with ac = aAp, and a branching probability of p = 0.2590,174)...

Fig. 5. Kratky plot for SANS data taken from solutions of starburst polyamidoamine dendrimers with seven generations in deuterated water (extrapolated to zero concentration) and comparison with the gaussian blocks dendrimer model with b = 0.48 nm, n = 10, N = 7, Nb = 3 and f = 2. Salt was added to the solution in order to screen out the coulomb interactions...

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