Abstract
Polyrotaxane (PR) possesses a supramolecular structure in which cyclic molecules are threaded into an axial polymer. In this study, the static structure of PR dissolved in a good solvent was investigated using contrast variation small-angle neutron scattering. The conformation of the axial linear polymer and the alignment of cyclic molecules within the axial polymer were evaluated quantitatively with the help of a detailed derivation of scattering theory. The decomposed partial scattering functions of the cyclic molecules and the axial polymer and the cross-correlation between cyclic molecules and an axial polymer strongly supported the idea that the alignment of cyclic molecules threaded on the axial polymer is random. On the basis of experimental observation, the entropic origin of the stiffening of PR due to the array of cyclic molecules is discussed.
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Acknowledgements
This work was partially supported by the Ministry of Education, Science, Sports and Culture, Japan (Grant-in-Aid for Scientific Research on Priority Areas, 2006–2010, No. 18068004 and Grant-in-Aid for Scientific Research (S), 2008–2012, No. 20221005). The SANS experiment was performed with the approval of the Institute for Solid State Physics, The University of Tokyo, at the Japan Atomic Energy Agency, Tokai, Japan (Proposal No. 7607).
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Appendix A
Appendix A
Form factor of a discrete rod
The scattering amplitude of a rod of length L is given by
where the classic coordinate system with the z axis in the direction of the vector Q (Q·r=Qr cosθ) is used. Then, an analytical form of the form factor of the rod can be obtained:
which is identical to Equation (5).
We discretize Equation (A1) by assuming that the rod consists of monomer-A and monomer-B, where the total number of monomers is n0 and the length of one monomer unit is L/n0. The discretization proceeds as follows:
Therefore, the corresponding form factor of a discrete rod composed of monomer-A is obtained by
with the n0-dimensional vector Am/n(k) defined by Equation (8). In the case of a purely random process for the distribution of monomer-A, Am/n(k) can be substituted by the ensemble average defined by Equation (9); then, an analytic form of Equation (A4) can be derived as
where PR(x)=2Si(x)/x−{sin(x)/x}2 with the sine integral Si(x).
In the case of a discrete cylinder with base radius R and length L, instead of the discrete rod, Equation (A4) is modified as
where PDisk(x)={2J1(x)/x}2 with the first-order cylindrical Bessel function J1(x). For hollow cylindrical objects such as cyclic molecules, PDisk(x) can be substituted by
where Rin is the internal radius and Rout is the external radius. For the purely random process of the distribution of monomer-A, Equation (A6) can be reduced to
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Endo, H., Mayumi, K., Osaka, N. et al. The static structure of polyrotaxane in solution investigated by contrast variation small-angle neutron scattering. Polym J 43, 155–163 (2011). https://doi.org/10.1038/pj.2010.124
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DOI: https://doi.org/10.1038/pj.2010.124
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