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Oscillating Free Induction Decay in Polymer Systems: Theoretical Analysis

  • Theory and Simulation
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Abstract

It is shown that during the Anderson-Weiss transformation derivation, as applied to a reptating polymer chain, the spin system can be divided into two sub-systems: an ergodic fluctuating one and a nonergodic quasi-static one. Consequently, the free induction decay (FID) expression is factorized. The factor with a fluctuating dipole-dipole interaction for an arbitrary correlation time is transformed into the generalized Anderson-Weiss exponential, and the factor with a quasi-static dipole-dipole interaction in an isotropic polymer melt is transformed into the oscillating Fourier image of the Pake doublet. The final expression permits both describing the FID shape of a polymer melt as a function of the molecular mass and temperature below the temperature of the primitive segment quasi-static state and forecasting the temperature range in which oscillating FIDs have been observed for poly(isoprene) melt. The new approach is applicable to qualitatively describe the oscillating FID in semi-crystalline poly(ethylene).

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References

  1. J.-P. Cohen-Addad, J. Chem. Phys. 60 (6), 2440 (1974).

    Article  CAS  Google Scholar 

  2. J.-P. Cohen-Addad, J. Phys. (Paris) 43 (10), 1509 (1982).

    Article  CAS  Google Scholar 

  3. V. D. Fedotov, V. M. Chernov, and T. N. Khazanovich, Polym. Sci. 20 (4), 1037 (1978).

    Google Scholar 

  4. Yu. Ya. Gotlib, M. I. Lifshits, V. A. Shevelev, I. S. Lishanskii, and I. V. Balanina, Polym. Sci. 18 (10), 2630 (1976).

    Google Scholar 

  5. R. Folland and A. Charlesby, Polymer 20 (2), 207 (1979).

    Article  CAS  Google Scholar 

  6. T. P. Kulagina, V. V. Marchenkov, and B. N. Provotorov, Polym. Sci. 31 (2), 420 (1989).

    Google Scholar 

  7. T. P. Kulagina, V. A. Varakina, and A. N. Kuzina, Dokl. Phys. Chem 444 (1), 79 (2012).

    Article  CAS  Google Scholar 

  8. M. G. Brereton, J. Chem. Phys. 94 (3), 2136 (1991).

    Article  CAS  Google Scholar 

  9. R. Kimmich, M. Kopf, and P. Callaghan, J. Polym. Sci., Part B: Polym. Phys. 29 (10), 1025 (1991).

    Article  CAS  Google Scholar 

  10. P. T. Callaghan, Polymer 29 (11), 1951 (1998).

    Article  Google Scholar 

  11. R. Kimmich and N. Fatkullin, Adv. Polym. Sci. 170 (2004).

  12. J. Collignon, H. Sillescu, and H. W. Spiess, Colloid Polym. Sci. 259 (3), 220 (1981).

    Article  CAS  Google Scholar 

  13. J.-P. Cohen-Addad and C. Schmit, Polymer 29 (5), 883 (1988).

    Article  CAS  Google Scholar 

  14. A. Guillermo, C. Lartigue, and J.-P. Cohen-Addad, Macromolecules 31 (3), 769 (1998).

    Article  CAS  Google Scholar 

  15. J.-P. Cohen-Addad and E. Guillermo, J. Chem. Phys. 111 (15), 7131 (1999).

    Article  CAS  Google Scholar 

  16. J.-P. Cohen-Addad, E. Guillermo, and C. Lartigue, Phys. Rev. Lett. 85 (16), 3432 (2000).

    Article  PubMed  CAS  Google Scholar 

  17. A. Guillermo, J.-P. Cohen-Addad, and D. Bytchenkoff, J. Chem. Phys. 113 (12), 5098 (2000).

    Article  CAS  Google Scholar 

  18. E. Guillermo and J.-P. Cohen-Addad, J. Chem. Phys. 116 (7), 3141 (2002).

    Article  CAS  Google Scholar 

  19. E. Schille, J.-P. Cohen-Addad, and E. Guillermo, Macromolecules 37 (6), 2144 (2004).

    Article  CAS  Google Scholar 

  20. N. Todica, Int. J. Mod. Phys. B 19 (10), 1771 (2005).

    Article  CAS  Google Scholar 

  21. A. Kimmich, E. Fischer, P. Callaghan, and N. Fatkullin, J. Magn. Reson., Ser. A 117 (1), 53 (1995).

    Article  CAS  Google Scholar 

  22. R. Graf, A. Heuer, and H. W. Spiess, Phys. Rev. Lett. 80 (26), 5738 (1998).

    Article  CAS  Google Scholar 

  23. F. V. Chavez and K. Saalwachter, Macromolecules 44 (6), 1549 (2011).

    Article  CAS  Google Scholar 

  24. A. Herrmann, B. Kresse, B. Gmeiner, A. F. Privalov, D. Kruk, D. Fujara, and E. A. Rossler, Macromolecules 45 (3), 1408 (2012).

    Article  CAS  Google Scholar 

  25. M. E. Hoffmann, A. Hermann, A. Abou Elfadl, A. Wohlfahrt, and E. A. Rossler, Macromolecules 45 (5), 2390 (2012).

    Article  CAS  Google Scholar 

  26. K. Saalwachter, Macromolecules 46 (12), 5090 (2013).

    Article  CAS  Google Scholar 

  27. K. Saalwachter, Prog. Nucl. Magn. Reson. Spectrosc. 51 (1), 1 (2007).

    Article  CAS  Google Scholar 

  28. B. Ekanayake, H. Menge, S. Korner, H. Schneider, M. E. Ries, and M. G. Brereton, Macromolecules 34 (14), 4683 (2001).

    Article  CAS  Google Scholar 

  29. B. Ekanayake, H. Menge, M. E. Rice, and M. G. Brereton, Macromolecules 35 (11), 4343 (2002).

    Article  CAS  Google Scholar 

  30. S. F. Schneider, S. F. Demco, and B. Blumich, Macromolecules 34 (12), 4019 (2001).

    Article  CAS  Google Scholar 

  31. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).

    Google Scholar 

  32. A. Yu. Grossberg and A. R. Khokhlov, Statistical Physics of Macromolecules (AIP, New York, 1994).

    Google Scholar 

  33. Yu. Ya. Gotlib, A. A. Darinskii, and Yu. E. Svetlov, Physical Kinetics of Macromolecules (Khimiya, Leningrad, 1986) [in Russian].

    Google Scholar 

  34. J. D. Schieber, J. Chem. Phys. 118 (1), 5162 (2003).

    Article  CAS  Google Scholar 

  35. C. Tzoumanekas and D. N. Theodorou, Macromolecules 39 (13), 4592 (2006).

    Article  CAS  Google Scholar 

  36. T. Spyriouni, C. Tzoumanekas, D. N. Theodorou, F. Muller-Plate, and G. Milano, Macromolecules 40 (10), 3876 (2007).

    Article  CAS  Google Scholar 

  37. C. Baig, P. S. Stephanou, G. Tsolou, V. G. Mavrantsas, and V. G. Kroger, Macromolecules 43 (19), 8239 (2010).

    Article  CAS  Google Scholar 

  38. M. Anagiannakis, C. Tzoumanekas, and D. N. Theodorou, Macromolecules 45 (23), 9475 (2012).

    Article  CAS  Google Scholar 

  39. K. Moldovan, R. Fechete, R. Demco, R. Culea, B. Blumich, B. Herrmann, and M. Heinz, J. Magn. Reson. 208 (1), 156 (2011).

    Article  PubMed  CAS  Google Scholar 

  40. A. Zhu, A. Huinink, P. C. M. M. Magusin, O. C. G. Adan, and K. Kopinga, J. Magn. Reson. 235 (10), 109 (2013).

    Article  PubMed  CAS  Google Scholar 

  41. G. E. Pake, J. Chem. Phys. 6 (4), 327 (1948).

    Article  Google Scholar 

  42. A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961).

    Google Scholar 

  43. E. Slichter, Principles of Magnetic Resonance (Springer, Berlin; New York, 1990).

    Book  Google Scholar 

  44. E. Sternin, M. Bloom, and A. L. MacKay, J. Magn. Reson. 55 (2), 274 (1983).

    CAS  Google Scholar 

  45. K_P. Whittall, E. Sternin, M. Bloom, and A. L. MacKay, J. Magn. Reson. 84 (1), 64 (1989).

    Google Scholar 

  46. M. A. McCabe and S. Wassal, Solid State Nucl. Magn. Reson. 10 (1) (1997).

    Google Scholar 

  47. K. Schafer, B. Madler, and E. Sternin, Biophys. J. 74 (2), 1007 (1998).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  48. V. Macho, L. Brombacher, and H. W. Spiess, Appl. Magn. Reson. 20 (3), 405 (2001).

    Article  CAS  Google Scholar 

  49. D. W. Elliot, W. P. Niemczura, and K. K. Kumashiro, J. Magn. Reson. 200 (1), 130 (2009).

    Article  CAS  Google Scholar 

  50. K. Hock and K. Earle, Appl. Magn. Reson. 45 (9), 859 (2014).

    Article  Google Scholar 

  51. G. Trausch, G. Canet, A. Cadene, and G. Turq, Chem. Phys. Lett. 433 (1–3), 228 (2006).

    Article  CAS  Google Scholar 

  52. G. Trausch and G. Canet, Magn. Reson. Imaging 25 (4), 525 (2007).

    Article  PubMed  CAS  Google Scholar 

  53. G. Trausch, G. Canet, and G. Turq, Chem. Phys. Lett. 456 (4–6), 262 (2008).

    Article  CAS  Google Scholar 

  54. J. A. Hassan, Phys. B (Amsterdam, Neth.) 407 (1), 179 (2012).

    Article  CAS  Google Scholar 

  55. S. E. Svanson, Acta Chem. Scand. 16 (9), 2212 (1962).

    Article  CAS  Google Scholar 

  56. R. A. Look, R. A. Lowe, and J. A. Northby, J. Chem. Phys. 44 (9), 3441 (1966).

    Article  CAS  Google Scholar 

  57. B. A. Van Baren, B. A. Emid, C. Steenbergen, and R. A. Wind, J. Magn. Reson. 4 (1), 466 (1972).

    Google Scholar 

  58. V. M. Chernov and G. S. Krasnopol’skii, J. Exp. Theor. Phys. 107 (2), 302 (2008).

    Article  CAS  Google Scholar 

  59. E. M. Pestryaev, J. Phys.: Conf. Ser. 1324 (1), 012031 (2011).

    Google Scholar 

  60. E. M. Pestryaev, Polym. Sci., Ser. A 55 (5), 336 (2013).

    CAS  Google Scholar 

  61. E. M. Pestryaev, Izv. Ufimskogo Nauch. Tsentra RAN, Ser. Phiz. 4 (3), 32 (2014).

    Google Scholar 

  62. E. M. Pestryaev, in Structure and Dynamics of Molecular Systems, Ed. by A. E. Chalykh (IPhKhE RAN, Moscow, 2016), Part 23, p. 378 [in Russian].

    Google Scholar 

  63. T. N. Khazanovich, Polym. Sci. 4 (4), 727 (1963).

    Google Scholar 

  64. V. P. Grigor’ev, A. I. Maklakov, and V. D. Skirda, Polym. Sci. 19 (2), 325 (1977).

    Google Scholar 

  65. J.-P. Cohen-Addad and R. Dupeyre, Polymer 24 (4), 400 (1983).

    Article  CAS  Google Scholar 

  66. J.-P. Cohen-Addad, Polymer 24 (9), 1128 (1983).

    Article  CAS  Google Scholar 

  67. H. Schneider and H. Hiller, J. Polym. Sci., Part B: Polym. Phys. 28 (7), 1001 (1990).

    Article  CAS  Google Scholar 

  68. F. V. Chavez and K. Saalwachter, Macromolecules 44 (6), 1560 (2011).

    Article  CAS  Google Scholar 

  69. M. E. Ries, A. Bansal, and M. G. Brereton, ACS Symp. Ser. 1077 (25), 417 (2011).

    Article  CAS  Google Scholar 

  70. S. Sturniolo and M. Pieruccini, J. Magn. Reson. 223 (11), 138 (2012).

    Article  PubMed  CAS  Google Scholar 

  71. S. Sturniolo, Sci. Acta 6 (1), 3 (2012).

    Google Scholar 

  72. S. Sturniolo, M. Pieruccini, M. Corti, and A. Rigamonti, Solid State Nucl. Magn. Reson. 5152 (45), 16 (2013).

    Article  CAS  Google Scholar 

  73. M. G. Brereton, Macromolecules 22 (9), 3667 (1989).

    Article  CAS  Google Scholar 

  74. M. G. Brereton, Macromolecules 23 (4), 1119 (1990).

    Article  CAS  Google Scholar 

  75. M. G. Brereton, I. M. Ward, N. Boden, and P. Wright, Macromolecules 24 (8), 2068 (1991).

    Article  CAS  Google Scholar 

  76. R. C. Ball, P. T. Callaghan, and E. T. Samulski, J. Chem. Phys. 106 (17), 7352 (1997).

    Article  CAS  Google Scholar 

  77. P. T. Callaghan and E. T. Samulski, Macromolecules 31 (11), 3693 (1998).

    Article  CAS  Google Scholar 

  78. T. P. Kulagina, P. S. Manikin, P. S. Karnauch, and L. P. Smirnov, Russ. J. Phys. Chem. 5 (4), 674 (2011).

    Article  CAS  Google Scholar 

  79. K. V. Fatkullin, K. V. Gubaidullin, K. V. Mattea, and S. Stapf, J. Chem. Phys. 137 (22), 224907 (2012).

    Article  PubMed  CAS  Google Scholar 

  80. K. V. Fenchenko, Polym. Sci., Ser. A 44 (6), 1029 (2002).

    Google Scholar 

  81. K. V. Fenchenko, J. Non-Cryst. Solids 358 (3), 474 (2012).

    Article  CAS  Google Scholar 

  82. J. Colmenero and A. Arbe, J. Polym. Sci., Part B: Polym. Phys. 51 (2), 87 (2013).

    Article  CAS  Google Scholar 

  83. N. Bloembergen, N. Purcell, and R. V. Pound, Phys. Rev. 73 (7), 679 (1948).

    Article  CAS  Google Scholar 

  84. R. J. A. Steenbakkers, R. J. A. Tzoumanekas, Y. Li, Y. Kroger, and J. D. Shieber, New J. Phys. 16 (1), 015027 (2014).

    Article  Google Scholar 

  85. P. A. Colmenero, A. J. Moreno, and A. J. Alegria, Prog. Polym. Sci. 30 (12), 1147 (2005).

    Article  CAS  Google Scholar 

  86. P. A. Beckmann, Solid State Nucl. Magn. Reson. 71 (10), 91 (2015).

    Article  PubMed  CAS  Google Scholar 

  87. Ye. M. Pestryayev and I. P. Storozhuk, Polym. Sci. 22 (9), 2309 (1980).

    Google Scholar 

  88. C. A. Steren and G. A. Monti, Macromolecules 37 (15), 5624 (2004).

    Article  CAS  Google Scholar 

  89. L. Wallach, J. Chem. Phys. 47 (12), 5258 (1967).

    Article  CAS  Google Scholar 

  90. L. Wittebort and L. Szabo, J. Chem. Phys. 69 (4), 1722 (1978).

    Article  CAS  Google Scholar 

  91. I. Sen, I. Bahar, I. Erman, I. Laupretre, and L. Monnerie, Macromolecules 32 (9), 3017 (1999).

    Article  CAS  Google Scholar 

  92. I. Bahar and I. Erman, Macromolecules 20 (9), 1368 (1987).

    Article  CAS  Google Scholar 

  93. G. D. Smith, G. D. Yoon, W. Zhu, and M. D. Ediger, Macromolecules 27 (20), 5563 (1994).

    Article  CAS  Google Scholar 

  94. G. D. Smith, G. D. Yoon, C. G. Wade, D. O’Leary, A. Chen, and R. L. Jaffe, J. Chem. Phys. 106 (9), 3798 (1996).

    Article  Google Scholar 

  95. A. A. Elfadl, R. Kahlau, A. Herrmann, V. N. Novikov, and E. A. Rossler, Macromolecules 43 (9), 3340 (2010).

    Article  CAS  Google Scholar 

  96. F. Furtado, J. Damron, M.-L. Trutscheel, C. Franz, K. Schreter, R. C. Ball, and K. Saalwachter, Macromolecules 47 (1), 256 (2014).

    Article  CAS  Google Scholar 

  97. L. J. Fetters, D. J. Lohse, D. Richter, T. A. Witten, and A. Zirkel, Macromolecules 27 (17), 4639 (1994).

    Article  CAS  Google Scholar 

  98. L. J. Fetters, D. J. Lohse, S. T. Milner, and W. W. Graessley, Macromolecules 32 (20), 6847 (1999).

    Article  CAS  Google Scholar 

  99. C. Riedel, A. Alegria, P. Tordjeman, and J. Colmenero, Rheol. Acta 49 (5), 507 (2010).

    Article  CAS  Google Scholar 

  100. P. Sotta and B. Deloche, Macromolecules 23 (7), 1999 (1990).

    Article  CAS  Google Scholar 

  101. L. Jarecki and A. Ziabicki, Polymer 43 (8), 2549 (2002).

    Article  CAS  Google Scholar 

  102. L. Jarecki and A. Ziabicki, Polymer 43 (14), 4063 (2002).

    Article  CAS  Google Scholar 

  103. A. Brodin, J. Chem. Phys. 128 (10), 104901 (2008).

    Article  PubMed  CAS  Google Scholar 

  104. V. M. Chernov and A. V. Butakov, Polym. Sci., Ser. A. 53 (3), 204 (2011).

    Article  CAS  Google Scholar 

  105. P. T. Callaghan and E. T. Samulski, Macromolecules 36 (3), 724 (2003).

    Article  CAS  Google Scholar 

  106. M. Febbo, V. Rostiashvilli, D. Dimitrov, and T. A. Vilgis, J. Chem. Phys. 129 (15), 154908 (2008).

    Article  PubMed  CAS  Google Scholar 

  107. R. T. Boere and R. G. Kidd, Annu. Rep. NMR Spectrosc. 13, 319 (1982).

    Article  CAS  Google Scholar 

  108. L. Latanowicz, Concepts Magn. Reson. 40A (2), 66 (2007).

    Article  CAS  Google Scholar 

  109. D. E. Woessner, eMagRes (2007). doi: 10.1002/9780470034590.emrstm0047

    Google Scholar 

  110. T. C. B. McLeish, Adv. Phys. 51 (6), 1379 (2002).

    Article  CAS  Google Scholar 

  111. A. E. Likhtman and T. C. B. McLeish, Macromolecules 35 (16), 6332 (2002).

    Article  CAS  Google Scholar 

  112. A. Drzewinski and J. M. J. Van Leeuwen, Phys. Rep. 475 (56), 53 (2009).

    Google Scholar 

  113. H. J. Unidad, M. A. G oad, A. R. Bras, M. Zamponi, R. Faust, J. Algaier, W. Pyckhout-Hintzen, A. Wischnewski, D. Richter, and L. J. Fetters, Macromolecules 48 (24), 6638 (2015).

    Article  CAS  Google Scholar 

  114. Connor T. M. // Trans. Faraday Soc. 60, 1574 (1964).

    Article  CAS  Google Scholar 

  115. V. J. McBrierty and D. C. Douglass, Phys. Rep. 63 (2), 61 (1980).

    Article  CAS  Google Scholar 

  116. P. A. Beckmann, Phys. Rep. 171 (3), 85 (1988).

    Article  Google Scholar 

  117. A. Bormuth, M. Henritzi, and M. Vogel, Macromolecules 43 (8), 8985 (2010).

    Article  CAS  Google Scholar 

  118. A. Bormuth, M. Hofman, M. Henritzi, M. Vogel, E. A. Rossler, Macromolecules 46 (19), 7805 (2013).

    Article  CAS  Google Scholar 

  119. P. P. Elfadl, R. Kahlau, A. Herrmann, V. N. Novikov, and E. A. Rossler, Macromolecules 43 (7), 3340 (2010).

    Article  CAS  Google Scholar 

  120. P. P. Kornfield and P. P. Chung, Macromolecules 25 (17), 4442 (1992).

    Article  CAS  Google Scholar 

  121. V. M. Litvinov, M. E. Ries, T. W. Baughman, A. Henke, and P. P. Matloka, Macromolecules 46 (2), 541 (2013).

    Article  CAS  Google Scholar 

  122. G. C. Borgia, R. J. S. Brown, and P. Fantazzini, J. Magn. Reson. 132 (1), 65 (1998).

    Article  PubMed  CAS  Google Scholar 

  123. P. J. Song, L. Venkataramanan, M. D. Hurlimann, M. Flaum, P. Frulla, and C. Straley, J. Magn. Reson. 154 (2), 261 (2006).

    Article  CAS  Google Scholar 

  124. C. G. Koay and P. J. Basser, J. Magn. Reson. 179 (2), 317 (2006).

    Article  PubMed  CAS  Google Scholar 

  125. S. Ghosh, K. M. Keener, and Y. Pan, J. Magn. Reson. 191 (2), 266 (2008).

    Article  CAS  Google Scholar 

  126. X. Li, L. Kong, J. Cheng, L. Wu, J. Geophys. Eng. 12 (1), 144 (2015).

    Article  Google Scholar 

  127. L. Dagys, V. Klimavicius, and V. Balevicius, J. Chem. Phys. 145 (11), 114202 (2016).

    Article  CAS  Google Scholar 

  128. R. Barenwald, Y. Champoret, K. Saalwachter, and K. Schaler, J. Phys. Chem. B 116 (25), 13089 (2012).

    Article  PubMed  CAS  Google Scholar 

  129. G. Williams, and D. C. Watts, Trans. Faraday Soc. 66, 80 (1970).

    Article  CAS  Google Scholar 

  130. G. B. Williams, D. C. Watts, S. B. Dev, A. M. North, Trans. Faraday Soc. 67, 1323 (1971).

    Article  CAS  Google Scholar 

  131. R. Lindsey, and G. B. Patterson, J. Chem. Phys. 73 (7), 3348 (1980).

    Article  CAS  Google Scholar 

  132. R. Zorn, J. Chem. Phys. 116 (8), 3204 (2002).

    Article  CAS  Google Scholar 

  133. M. N. Berberan-Santos, E. N. Bodunov, and B. Valeur, Chem. Phys. 315 (12), 171 (2005).

    Article  CAS  Google Scholar 

  134. D. C. Johnston, Phys. Rev. B 74 (18), 184443 (2006).

    Article  CAS  Google Scholar 

  135. U. Powles, D. M. Heyes, G. Rickayzen, and W. A. B. Evans, J. Chem. Phys. 131 (21), 214509 (2009).

    Article  PubMed  CAS  Google Scholar 

  136. R. Garrapa, F. Mainardi, and G. Maione, Fractional Calculus Appl. Anal. 19 (5), 1105 (2016).

    Google Scholar 

  137. X. Gong, E. W. Hansen, and Q. Chen, Macromol. Chem. Phys. 212 (10), 1007 (2011).

    Article  CAS  Google Scholar 

  138. X. Gong, E. W. Hansen, and Q. Chen, Macromol. Chem. Phys. 213 (3), 278 (2012).

    Article  CAS  Google Scholar 

  139. X. Gong, E. W. Hansen, and Q. Chen, Macromol. Chem. Phys. 213 (23), 2464 (2012).

    Article  CAS  Google Scholar 

  140. E. W. Hansen, X. Gong, and Q. Chen, Macromol. Chem. Phys. 214 (7), 844 (2013).

    Article  CAS  Google Scholar 

  141. W. Weibull, J. Appl. Mech. 18 (9), 293 (1951).

    Google Scholar 

  142. S. Kaufman and D. J. Bunger, J. Magn. Reson. 2 (7), 218 (1970).

    Google Scholar 

  143. P. W. Anderson and P. R. Weiss, Rev. Mod. Phys. 25 (1), 269 (1953).

    Article  CAS  Google Scholar 

  144. P. W. Anderson, J. Phys. Soc. Jpn. 9 (3), 316 (1954).

    Article  Google Scholar 

  145. D. E. Woessner, Concepts Magn. Reson. 8 (6), 397 (1996).

    Article  CAS  Google Scholar 

  146. E. R. Andrew, J. Chem. Phys. 18 (5), 607 (1950).

    Article  CAS  Google Scholar 

  147. E. R. Andrew and R. G. Eades, Proc. R. Soc. London, Ser. A 218 (1135), 537 (1953).

    Article  CAS  Google Scholar 

  148. H. G. Olf and A. Peterlin, Kolloid Z. Z. Polym. 215 (2), 97 (1967).

    Article  CAS  Google Scholar 

  149. H. G. Olf and A. Peterlin, J. Appl. Phys. 35 (11), 3108 (1964).

    Article  CAS  Google Scholar 

  150. H. G. Olf and A. Peterlin, J. Polym. Sci., Part A-2 8 (5), 753 (1970).

    Article  CAS  Google Scholar 

  151. H. G. Olf and A. Peterlin, J. Polym. Sci., Part A-2 8 (5), 771 (1970).

    Article  CAS  Google Scholar 

  152. H. G. Olf and A. Peterlin, J. Polym. Sci., Part A-2 8 (5), 791 (1970).

    Article  CAS  Google Scholar 

  153. M. Polak, M. Sheinblatt, and U. Shmueli, J. Magn. Reson. 16 (2), 252 (1974).

    CAS  Google Scholar 

  154. H. Shmueli, M. Polak, and M. Sheinblatt, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 32 (2), 192 (1976).

    Article  Google Scholar 

  155. J. Collignon and H. Sillescu, J. Polym. Sci., Polym. Lett. Ed. 18 (10), 669 (1980).

    Article  CAS  Google Scholar 

  156. N. Lindner, E. Rossler, and H. Sillescu, Makromol. Chem. 182 (12), 3653 (1981).

    Article  CAS  Google Scholar 

  157. R. Fischer, R. Kimmich, and N. Fatkullin, J. Chem. Phys. 106 (23), 9883 (1997).

    Article  CAS  Google Scholar 

  158. M. Kehr, M. Fatkullin, and R. Kimmich, J. Chem. Phys. 127 (8), 084911 (2007).

    Article  PubMed  CAS  Google Scholar 

  159. N. Fatkullin, A. Gubaidullin, and S. Stapf, J. Chem. Phys. 132 (9), 094903 (2010).

    Article  PubMed  CAS  Google Scholar 

  160. B. Kruk, R. Meier, and E. A. Rossler, Phys. Rev. E 85 (2), 020201(R) (2012).

    CAS  Google Scholar 

  161. Kruk B., Hermann B., Rossler E. A., Prog. Nucl. Magn. Reson. Spectrosc. 63 (3), 33 (2012).

    Article  PubMed  CAS  Google Scholar 

  162. B. Kresse, B. Privalov, and B. Fujara, Solid. State Magn. Reson. 40 (4), 134 (2011).

    Article  CAS  Google Scholar 

  163. B. Kresse, B. Becher, A. F. Privalov, M. Hofmann, E. A. Rossler, M. Vogel, and F. Fujara, J. Magn. Reson. 277 (4), 79 (2017).

    Article  PubMed  CAS  Google Scholar 

  164. F. Fujara, D. Kruk, and A. F. Privalov, Progr. Nucl. Magn. Reson. Spectrsc. 82 (2), 39 (2014).

    Article  CAS  Google Scholar 

  165. A. Meier, D. Kruk, E. A. Rossler, ChemPhysChem 14 (3), 3071(2013).

    Article  PubMed  CAS  Google Scholar 

  166. R. Meier, A. Hermann, B. Kresse, A. F. Privalov, D. Kruk, F. Fujara, and E. A. Rossler, ACS Macro Lett. 2 (2), 96 (2013).

    Article  CAS  Google Scholar 

  167. A. Fatkullin, S. Stapf, M. Hofmann, R. Meier, and E. A. Rossler, J. Non-Cryst. Solids 407 (1), 309 (2015).

    Article  CAS  Google Scholar 

  168. R. Meier, D. Kruk, A. Bourdick, E. Schneider, and E. A. Rossler, Appl. Magn. Reson. 44 (12), 153 (2013).

    Article  CAS  Google Scholar 

  169. B. Kresse, M. Hofmann, A. F. Privalov, N. Fatkullin, F. Fujara, and E. A. Rossler, Macromolecules 48 (13), 4491(2015).

    Article  CAS  Google Scholar 

  170. M. Flamig, M. Becher, M. Hofmann, T. Korber, B. Kresse, A. F. Privalov, L. Willner, D. Kruk, F. Fujara, and E. A. Rossler, J. Phys. Chem. 120 (31), 7754 (2016).

    Article  CAS  Google Scholar 

  171. C. Lartigue, A. Guillermo, and J.-P. Cohen-Addad, J. Polym. Sci., Part B: Polym. Phys. 35 (7), 1095 (1997).

    Article  CAS  Google Scholar 

  172. G. Simon, K. Baumann, and W. Gronski, Macromolecules 25 (14), 3624 (1992).

    Article  CAS  Google Scholar 

  173. M. Hofmann, B. Kresse, L. Heymann, A. F. Privalov, L. Willner, N. Fatkullin, N. Aksel, F. Fujara, and E. A. Rossler, Macromolecules 49 (22), 8622 (2016).

    Article  CAS  Google Scholar 

  174. P. Henritzi, A. Bormuth, and M. Vogel, Solid State Nucl. Magn. Reson. 54 (9), 32 (2013).

    Article  PubMed  CAS  Google Scholar 

  175. C. A. Jeener and P. Broekaert, Phys. Rev. 157 (2), 232 (1967).

    Article  CAS  Google Scholar 

  176. P. Sotta and C. A. Deloche, J. Chem. Phys. 100 (6), 4591 (1994).

    Article  CAS  Google Scholar 

  177. C. A. Sholl, J. Phys. C: Solid State Phys. 14. (4), 447 (1981).

    Article  CAS  Google Scholar 

  178. C. A. Sholl, J. Phys.: Condens. Matter 12 (18), 4285 (2000).

    CAS  Google Scholar 

  179. M. F. Herman, J. Chem. Phys. 112 (6), 3040 (2000).

    Article  CAS  Google Scholar 

  180. Z. Wang, A. E. Likhtman, R. G. Larson, Macromolecules 45 (8), 3557 (2012).

    Article  CAS  Google Scholar 

  181. A. Herrmann, V. N. Novikov, and E. A. Rossler, Macromolecules 42 (6), 2063 (2009).

    Article  CAS  Google Scholar 

  182. M. Roos, M. Ott, M. Hofmann, S. Link, E. A. Rossler, J. Balbach, A. Krushelnitsky, and K. Saalwachter, J. Am. Chem. Soc. 138 (32), 10365 (2016).

    Article  PubMed  CAS  Google Scholar 

  183. E. M. Pestryaev, V. D. Skirda, and A. I. Maklakov, Vysokomol. Soedin., Ser. B 23 (1), 3 (1981).

    CAS  Google Scholar 

  184. L. P. Powles and J. H. Strange, Proc. Phys. Soc., London 82 (1), 6 (1963).

    Article  CAS  Google Scholar 

  185. L. P. Hwang and L. P. Freed, J. Chem. Phys. 63 (9), 4017 (1975).

    Article  CAS  Google Scholar 

  186. E. Belorizky, D. G. Gillis, W. Gorecki, K. Lang, F. Noak, C. Roux, J. Struppe, L. H. Sutcliffe, J. P. Travers, and X. Wu, J. Phys. Chem. A 102. (21), 3674 (1998).

    Article  CAS  Google Scholar 

  187. K. Binder and A. P. Young, Rev. Mod. Phys. 58 (4), 801 (1986).

    Article  CAS  Google Scholar 

  188. M. Dzugutov, J. Phys.: Condens. Matter 11 (10), A253 (1999).

    CAS  Google Scholar 

  189. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford Univ. Press, Oxford, 2001).

    Google Scholar 

  190. K. Binder, J. Baschnagel, and W. Paul, Prog. Polym. Sci. 28 (1), 115 (2003).

    Article  CAS  Google Scholar 

  191. A. Gubaidullin, T. Shakirov, N. Fatkullin, and R. Kimmich, Solid State Nucl. Magn. Reson. 35 (3), 147 (2009).

    Article  PubMed  CAS  Google Scholar 

  192. G. Lipari and A. Szabo, Biophys J. 30. (3), 489 (1980).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  193. G. Lipari and A. Szabo, J. Am. Chem. Soc. 104 (17), 4546 (1982).

    Article  CAS  Google Scholar 

  194. G. Lipari and A. Szabo, J. Am. Chem. Soc. 104 (17), 4559 (1982).

    Article  CAS  Google Scholar 

  195. B. Halle and H. Wennerstrom, J. Chem. Phys. 75 (4), 1928 (1981).

    Article  CAS  Google Scholar 

  196. B. Halle, J. Chem. Phys. 131 (22), 224507 (2009).

    Article  PubMed  CAS  Google Scholar 

  197. E. Meirovich, Yu. E. Shapiro, A. Polimeno, and J. H. Freed, Prog. Nucl. Magn. Reson. Spectrosc. 56 (4), 360 (2010).

    Article  CAS  Google Scholar 

  198. D. E. Woessner and D. E. Snowden, Jr., J. Chem. Phys. 50 (4), 1516 (1969).

    Article  CAS  Google Scholar 

  199. S. Gade and I. J. Lowe, Phys. Rev. 148 (1), 382 (1966).

    Article  CAS  Google Scholar 

  200. S. Gade, Phys. Rev. 187 (2), 419 (1969).

    Article  CAS  Google Scholar 

  201. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

    Google Scholar 

  202. M. A. Heald, Math. Comput. 44 (170), 459 (1985).

    Article  Google Scholar 

  203. K. J. Fritzsching, K. Mao, and K. Schmidt-Rohr, Macromolecules 50 (4), 1521 (2017).

    Article  CAS  Google Scholar 

  204. D. W. McCall and D. C. Douglass, Polymer 4, 433 (1963).

    Article  CAS  Google Scholar 

  205. E. W. Hansen, P. E. Kristiansen, and B. Pedersen, J. Phys. Chem. B 102 (28), 5444 (1998).

    Article  CAS  Google Scholar 

  206. D. Dadayli, R. K. Harris, A. M. Kenwright, B. J. Say, and M. Sunnetcioglu, Polymer 35 (19), 4083 (1994).

    Article  CAS  Google Scholar 

  207. W. P. Weglarz, H. Peemoeller, and A. Rudin, J. Polym. Sci., Part B: Polym. Phys. 38 (19), 2487 (2000).

    Article  CAS  Google Scholar 

  208. V. D. Fedotov and A. N. Abdrashitova, Polym. Sci. 22 (3), 688 (1980).

    Google Scholar 

  209. A. A. Lundin and A. V. Makarenko, Chem. Phys. Lett. 83 (1), 192 (1981).

    Article  CAS  Google Scholar 

  210. H. Uehara, K. Yamanobe, and T. Komoto, Macromolecules 33 (13), 4861 (2000).

    Article  CAS  Google Scholar 

  211. T. Yamanobe, H. Uehara, and M. Kakiage, Annu. Rep. NMR Spectrosc. 70, 203 (2010).

    Article  CAS  Google Scholar 

  212. R. Barenwald, S. Goerlitz, R. Godehardt, A. Osichow, Q. Tong, M. Krumova, S. Mecking, and K. Saalwachter, Macromolecules 47 (15), 5163 (2014).

    Article  CAS  Google Scholar 

  213. W. Derbyshire, M. Van den Bosch, D. Van Dusschoten, W. MacNaughtan, I. A. Farhat, M. A. Hemminga, and J. R. Mitchell, J. Magn. Reson. 168 (2), 278 (2004).

    Article  PubMed  CAS  Google Scholar 

  214. J.-P. Douliez, A. Leonard, and E. Dufourc, Biophys. J. 68 (5), 1727 (1995).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  215. A. Vogel, C. P. Katzka, H. Waldmann, K. Arnold, M. F. Brown, and D. Huster, J. Am. Chem. Soc. 127 (5), 12263 (2005).

    Article  PubMed  CAS  Google Scholar 

  216. A. Vogel, K.-T. Tan, H. Waldmann, S. E. Feller, M. F. Brown, and D. Huster, Biophys. J. 93 (8), 2697 (2007).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  217. K. R. Jeffrey, T. C. Wong, A. P. Tulloch, Mol. Phys. 52 (2), 289 (1984).

    Article  CAS  Google Scholar 

  218. M. P. Gent and J. H. Prestegard, J. Magn. Reson. 25 (2), 243 (1977).

    CAS  Google Scholar 

  219. J. H. Davis and K. R. Jeffrey, Chem. Phys. Lipids 20 (2), 87 (1977).

    Article  CAS  Google Scholar 

  220. P. C. Dave, E. K. Tiburu, N. A. Nusair, and G. A. Lorigan, Solid State Nucl. Magn. Reson. 24 (23), 137 (2003).

    Article  PubMed  CAS  Google Scholar 

  221. K. Witte, B. E. S. Olausson, A. Walrant, I. D. Alves, and A. Vogel, Biochim. Biophys. Acta 1828 (2), 824 (2013).

    Article  PubMed  CAS  Google Scholar 

  222. S. Stahlberg, S. Lange, B. Dobner, and D. Huster, Langmuir 32 (8), 2023 (2016).

    Article  PubMed  CAS  Google Scholar 

  223. M. Shaghaghi, M.-T. Chen, Y.-W. Hsue, M. J. Zukermann, and J. L. Thewalt, Langmuir 32 (30), 7654 (2016).

    Article  PubMed  CAS  Google Scholar 

  224. S. Hayashi and Y. Komori, Solid State Nucl. Magn. Reson. 36 (4), 167 (2009).

    Article  PubMed  CAS  Google Scholar 

  225. R. Meier, Kahlau, and E. A. Rossler, J. Phys. Chem. A. 114 (30), 7847 (2010).

    Article  PubMed  CAS  Google Scholar 

  226. K. Adachi and T. Kotaka, Prog. Polym. Sci. 18 (3), 585 (1993).

    Article  CAS  Google Scholar 

  227. H. Watanabe, Prog. Polym. Sci. 24 (9), 1253 (1999).

    Article  CAS  Google Scholar 

  228. F. Kremer and A. Schonhals, Broadband Dielectric Spectroscopy (Springer, Berlin, 2003).

    Book  Google Scholar 

  229. K. Adachi and T. Kotaka, Macromolecules 18 (3), 466 (1985).

    Article  CAS  Google Scholar 

  230. S. Cerveny, P. Zink, M. Terrier, S. Arrese-Igor, A. Alegria, and J. Colmenero, Macromolecules 41 (22), 8669 (2008).

    Article  CAS  Google Scholar 

  231. J. Jackle, Rep. Prog. Phys. 49 (2), 171 (1986).

    Article  Google Scholar 

  232. J. M. Widmaier G. C. Meyer, Macromolecules 14 (2), 450 (1981).

    Article  CAS  Google Scholar 

  233. X. Lu and B. Jiang, Polymer 32 (3), 471 (1991).

    Article  CAS  Google Scholar 

  234. E. N. Datal, K. D. Taylor, and P. J. Phillips, Polymer 24 (12), 1623 (1983).

    Article  Google Scholar 

  235. R. E. Hammet, R. E. Wingard, and J. E. Phillips, Rubber Chem. Technol. 39 (2), 206 (1965).

    Article  Google Scholar 

  236. M. Shen, W. N. Hansen, and P. C. Romo, J. Chem. Phys. 51 (1), 425 (1969).

    Article  CAS  Google Scholar 

  237. S. Kavesh and J. M. Schultz, J. Polym. Sci., Part A-2 8 (2), 243 (1970).

    Article  CAS  Google Scholar 

  238. K. Iohara, K. Imada, and M. Takayanagi, Polym. J. 3 (3), 357 (1984).

    Article  Google Scholar 

  239. T. Kakudate, M. Kakizaki, and T. Hideshima, J. Polym. Sci., Part B: Polym. Phys. 23 (4), 787 (1985).

    CAS  Google Scholar 

  240. V. I. Sultanov, V. V. Atrazhev, D. V. Dmitriev, and S. F. Burlatsky, J. Exp. Theor. Phys. Lett. 98 (5), 294 (2013).

    Article  CAS  Google Scholar 

  241. K. Kuwabara, H. Kaji, M. Tsuji, and F. Horii, Macromolecules 33 (19), 7093 (2000).

    Article  CAS  Google Scholar 

  242. K. Schmidt-Rohr and H. W. Spiess, Macromolecules 24 (19), 5288 (1991).

    Article  CAS  Google Scholar 

  243. K. J. Zubova, N. K. Balabaev, and L. I. Manevitch, Polymer 48 (6), 1802 (2007).

    Article  CAS  Google Scholar 

  244. M. B. Robertson, I. M. Ward, P. G. Klein, and K. J. Packer, Macromolecules 30, 6893 (1997).

    Article  CAS  Google Scholar 

  245. L. Wang, X. Gao, Z. Sun, and J. Fenga, J. Chem. Phys. 130 (18), 184709 (2009).

    Article  PubMed  CAS  Google Scholar 

  246. L. Mandelkern, Rubber Chem. Technol. 32 (5), 1392 (1959).

    Article  Google Scholar 

  247. I. I. Perepechko, Introduction into Polymer Physics (Khimiya, Moscow, 1978) [in Russian].

    Google Scholar 

  248. D. W. Van Krevelen, Properties of Polymers, Ed. by K. te Nijenhuis (Elsevier, Amsterdam, 2009).

  249. C. Lorthior, S. Randriamahefa, and B. Deloche, J. Chem. Phys. 139 (22), 224903 (2013).

    Article  CAS  Google Scholar 

  250. C. Lorthior and B. Deloche, Colloid Polym. Sci. 292 (8), 1841 (2014).

    Article  CAS  Google Scholar 

  251. M. Z. Slimani, A. J. Moreno, and J. Colmenero, Macromolecules 44 (17), 6952 (2016).

    Article  CAS  Google Scholar 

  252. P. M. Singer, D. Asthagiri, W. G. Chapman, and G. J. Hiraski, J. Magn. Reson. 277 (4), 15 (2017).

    Article  PubMed  CAS  Google Scholar 

  253. R. E. Dehl, J. Chem. Phys. 60 (1), 339 (1974).

    Article  CAS  Google Scholar 

  254. D. Hentschel, H. Sillescu, and W. Spiess, Polymer 25 (8), 1078 (1984).

    Article  CAS  Google Scholar 

  255. J. McBrierty and I. M. Ward, J. Phys. D: Appl. Phys. 1 (1), 1529 (1968).

    Article  Google Scholar 

  256. M. J. Folkes and I. M. Ward, J. Mater. Sci. 6 (6), 582 (1971).

    Article  CAS  Google Scholar 

  257. I. Ya. Slonim, Russ. Chem. Rev. 31 (5), 609 (1962).

    Article  CAS  Google Scholar 

  258. G. Roudaut, I. Farhat, F. Poirer, and D. Champion, Carbohydr. Polym. 77 (3), 489 (2009).

    Article  CAS  Google Scholar 

  259. R. Voekel and H. Sillescu, J. Magn. Reson. 34 (3), 559 (1979).

    Google Scholar 

  260. W.-Y. Lin and F. D. Blum, Macromolecules 30 (18), 5331 (1997).

    Article  CAS  Google Scholar 

  261. W.-Y. Lin and F. D. Blum, Macromolecules 31 (13), 4135 (1998).

    Article  CAS  Google Scholar 

  262. T. Korber, F. Mohamed, M. Hofmann, A. Lichtinger, L. Wilner, and E. A. Rossler, Macromolecules 50 (4), 1554 (2017).

    Article  CAS  Google Scholar 

  263. H. W. Spiess, Colloid Polym. Sci. 261 (3), 193 (1983).

    Article  CAS  Google Scholar 

  264. H. W. Spiess, Adv. Polym. Sci. 66, 23 (1985).

    Article  CAS  Google Scholar 

  265. C. R. Bowers, Y. Wei, B. S. Aitken, C. R. Reeg, C. D. Akel, and K. B. Wagener, Polymer 53 (13), 2633 (2012).

    Article  CAS  Google Scholar 

  266. Q. Wulin, J. C. Sworen, M. Pyda, A. Habenschuss, K. B. Wagener, and B. Wunderlich, Macromolecules 39 (1), 204 (2006).

    Article  CAS  Google Scholar 

  267. Y. Wei, R. Graf, J. C. Sworen, C.-Y. Cheng, C. R. Bowers, K. B. Wagener, and H. W. Spiess, Angew. Chem., Int. Ed. 48 (25), 4617 (2009).

    Article  CAS  Google Scholar 

  268. L. C. Da Silva, R. Graf, C. R. Bowers, and K. B. Wagener, Macromolecules 48 (24), 8858 (2015).

    Article  CAS  Google Scholar 

  269. L. C. Da Silva, R. Graf, C. R. Bowers, and K. B. Wagener, Macromol. Rapid Commun. 37 (6), 527 (2016).

    Article  PubMed  CAS  Google Scholar 

  270. V. I. Arnold, Catastrophe Theory (Springer, Berlin, 2004).

    Google Scholar 

  271. S. P. Das, Rev. Mod. Phys. 76 (3), 785 (2004).

    Article  CAS  Google Scholar 

  272. K. Kim and S. Saito, J. Chem. Phys. 138 (12), 12A506 (2013).

    Article  PubMed  CAS  Google Scholar 

  273. E. M. Pestryaev, Izv. Ufimskogo Nauch. Tsentra RAN, Ser. Phiz. 4 (4), 54 (2014). http://elibrary.ru/contents.asp?issueid=1442986.

    Google Scholar 

  274. C. M. Roland, S. Hensel-Bielowka, M. Poluch, and R. Casalini, Rep. Prog. Phys. 68 (6), 1405 (2005).

    Article  CAS  Google Scholar 

  275. H. A. Resing, J. Chem. Phys. 43 (2), 669 (1965).

    Article  CAS  Google Scholar 

  276. L. J. Lynch, K. H. Marsden, and E. P. Geoge, J. Chem. Phys. 51 (12), 5673 (1969).

    Article  CAS  Google Scholar 

  277. L. J. Lynch, and K. H. Marsden, J. Chem. Phys. 51 (12), 5681 (1969).

    Article  PubMed  CAS  Google Scholar 

  278. K. Overloop, and L. Van Gerven, J. Magn. Reson., Ser. A 101 (2), 179 (1993).

    Article  CAS  Google Scholar 

  279. T. P. Kulagina, V. M. Litvinov, and K. T. Summanen, J. Polym. Sci., Part B: Polym. Phys. 31 (3), 241 (1993).

    Article  Google Scholar 

  280. N. Gelman and R. F. Code, Solid State Nucl. Magn. Reson. 14 (34), 191 (1999).

    Article  PubMed  CAS  Google Scholar 

  281. M. J. Duer, J. Magn. Reson. 253 (4), 98 (2015).

    Article  PubMed  CAS  Google Scholar 

  282. Yu. B. Grunin, L. Yu. Grunin, E. A. Nikolskaya, and V. I. Talantsev, Polym. Sci., Ser. A 54 (3), 201 (2012).

    CAS  Google Scholar 

  283. L. Yu. Grunin, Yu. B. Grunin, V. I. Talantsev, E. A. Nikolskaya, and D. S. Masas, Polym. Sci., Ser. A. 57 (1), 43 (2015).

    CAS  Google Scholar 

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    E. M. Pestryaev

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Original Russian Text © E.M. Pestryaev, 2018, published in Vysokomolekulyarnye Soedineniya, Seriya A, 2018, Vol. 60, No. 4, pp. 329–354.

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Pestryaev, E.M. Oscillating Free Induction Decay in Polymer Systems: Theoretical Analysis. Polym. Sci. Ser. A 60, 530–551 (2018). https://doi.org/10.1134/S0965545X18040090

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