Scattering function for wormlike chains with finite thickness

Abstract

The particle scattering function P(k) is approximately evaluated for the Kratky-Porod wormlike chain with a circular cross section to examine the effect of chain diameter d on the scattering curve of k 2 P(k) versus k, the magnitude of the scattering vector, for stiff chains and also the applicability of the cross-section plot of ln[kP(k)] versus k 2 to them. In the evaluation, series expansions from the rod and coil limits up to the fifth-order and third-order deviations, respectively, are combined together. The major results or conclusions derived are as follows. First, the conventional equation, P(k) ϭ P 0 (k) exp(Ϫk 2 d 2 /16), for straight cylinders overestimates k 2 P(k) at relatively large values of k, whereas its alternative, P(k) ϭ P 0 (k)[2J 1 (kd/2)/(kd/2)] 2 , is a good approximation to exact P(k) unless contour length L is shorter than 10d. Here, P 0 (k) denotes the scattering function for the chain contour, and J 1 (x) is the Bessel function of the first order. Second, as d is increased for a fixed value of L (relative to the Kuhn segment length), the k 2 P(k)-k curve lowers with a pronounced maximum, and the peak position shifts to a lower scattering angle. Third, if the chain is somewhat flexible, the cross-section plot has an approximately linear region, with a slope fairly close to Ϫd 2 /16 expected from the aforementioned conventional equation. This plot for rods appreciably bends down, and thus the experimental observation of an approximately linear relation (over a wide k range) implies that, in contrast to the prevailing notion, the polymer examined is not completely rigid but instead is somewhat flexible.

Surface and Interface Analyses of Polymer Brushes by Synchrotron Radiation

“…We estimated the radius of gyration of the polymer layer, s, of SiNP-PMAPS and SiNP-PMPC from the SAXS profile using a core-shell model analysis, taking the interacting excluded-volume chains into consideration 23) and assuming the corona chains as infinitely thin perturbed wormlike chains. 24) Figure 4 shows the C s -dependence of s and R H,brush for SiNP-PMAPS ( ) and SiNP-PMPC ( ) in aqueous NaCl solutions, where R H,brush were obtained by DLS measurements. R H,brush and s for SiNP-PMPC were constant over the entire C s region from 0 to 5.0 M. The C s -dependence of R H,brush for SiNP-PMPC is in good agreement with that reported by Matsuda and co-workers, who also did not observe a swelling of the PMPC brush upon the addition of up to 0.5 M of NaCl.…”

Solution Properties of Amylose Tris(Phenylcarbamate): Local Conformation and Chain Stiffness in 1,4-Dioxane and 2-Ethoxyethanol

“…Nakamura and Norisuye's theory 36 for PðkÞ of an unperturbed wormlike cylinder was applied to the SAXS data À1 values for ATPC50K; we note that ! À1 cannot be determined for ATPC20K because the theoretical solid lines fitting the data points in the two lower panels of the figure are hardly distinguishable from the corresponding dashed lines for the rod limit.…”

Small-angle X-ray scattering from polystyrene polymacromonomers with relatively short main chains

“…The dashed line for each plot shows the theoretical values for the cylindrical wormlike chain (Nakamura & Norisuye, 2004), where d values summarized in Table 3 were chosen to give the closest fit to the experimental data. These values were determined within AE3% and are about 10% smaller than the d values estimated from hS 2 i z in Table 2.…”