Self-diffusion in polymer systems, measured with field-gradient spin echo NMR methods

  • Ernst D. von Meerwall
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 54)


The steady gradient and pulsed gradient spin echo NMR Methods of measuring self-diffusion have for some twenty years been applied to the study of polymers. The methods are briefly described, and the principal results of this research are reviewed in three main areas: diffusion of polymers in the melt and in concentrated solutions, diffusion of polymer in dilute and semidilute solutions, and diffusion of penetrants and diluents in high polymers hosts. The theoretical interpretations of these experiments are included in the review, with particular attention to theories of dilute polymer solutions, the free-volume theory of diffusion in concentrated solutions, and power-law behavior postulated for various regimes.

The aim of this review is to familiarize workers in the polymer field with these techniques for measuring self-diffusion and with their applications and benefits.


Nuclear Magnetic Resonance Free Volume Polymer System Magnetic Field Gradient Solvent Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols and Abbreviations


height of spin echo in pulsed NMR, measured as function of field gradient parameters or time.


Solvent power parameter entering Flory's theory of dilute solutions.


degree of neutralization in polyelectrolyte solutions.


free-volume parameter entering Vrentas-Duda theory; subscript (1,2,i) denotes molecular species in solution.


ratio of vacancy size needed for diluent diffusion to diluent size, in the Fujita-Doolittle theory.


polymer concentration in solutions (mass/solution volume).

c*, c**

lower and upper limits of c in semidilute regime to which scaling laws apply.

C1, C2

parameters entering the Williams-Landel-Ferry equation.


referring to center-of-mass motion of a macromolecule.


magnetogyric ratio of nucleus at resonance (frequency/field).


diffusion distance.


diffusion coefficient; subscript indicates species diffusing.

D0, Di0

diffusion coefficient of a given species (i) in the limit of zero concentration in solution.

D0, D

diffusion coefficients measured at very short and very long diffusion times t.

Dmax, Dmin

diffusion coefficients measured at very small and very large values of gradient parameter δ2G2 at fixed diffusion time.


Cooperative diffusion coefficient.


duration of magnetic field gradient pulse.


interval between (beginning of) magnetic field gradient pulses.


thermal expansivity of free volume above the glass transition.


activation energy for self-diffusion.


viscosity of solvent.


fractional free volume; subscript (i, 1, 2, p, dil) denotes molecular species in solution.


fractional free volume at and below the glass transition.


functionality, number of arms of a star-branched polymer.


Field gradient spin echo method of measuring self-diffusion, encompasses SGSE and PGSE variants.


Fourier transform; refers to NMR experiments in which the time domain response of the spin system is transformed into a frequency spectrum.


overlap factor; describes multiple access to hole free volume for molecular transport.


magnitude of pulsed magnetic field gradient.


magnitude of steady magnetic field gradient.


Boltzmann constant


lowest-order coefficient of concentration dependence of 1/D in dilute polymer solutions.


extension ratio.

M, Mc

molecular weight; at the onset of entanglements.

\(\bar M_n ,\bar M_w \)

number average and weight average molecular weight of polydisperse polymer.


collision-dynamic mass entering the Vrentas-Duda theory; subscript indicates molecular species in solution.


exponent of molecular weight in scaling laws.


number of main-chain carbon atoms in polymer molecules.


nuclear magnetic resonance, pulsed or continuous-wave (CW).


pulsed-gradient spin echo method of measuring self-diffusion.


radio frequency; refers to NMR spectroscopy.


universal gas constant.


density (mass/volume).


steady-gradient spin-echo method of measuring self-diffusion.


mean squared linear dimension of coiled macromolecule in solution or in melt.


diffusion time.


time between radio frequency pulse and onset of gradient pulse (PGSE).


(absolute) temperature.

Tg, Tg∞

glass transition temperature; at infinite molecular weight.

T1, T

nuclear spin lattice relaxation time; in the rotating frame.


nuclear spin-spin relaxation time.


time between 90° and 180° rf pulses in T2 or FGSE experiment.


volume fraction; subscript (1, 2, i, p, dil) denotes species in solution.


molar free volume contributed by the ends of chain molecules.


molar free volume from any source.

\(\hat V_i^* \)

specific critical volume for diffusion of one molecule or segment of species i.


weight fraction of species i in solution.


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9 References

  1. 1.
    For recent reviews of the pertinent literature, see V. J. McBrierty and D. C. Douglass: Phys. Reports (Phys. Letters) 63, 61 (1980); also J. Polym. Sci. Macromol. Revs. 16, 295 (1981)Google Scholar
  2. 2.
    Slonim, I. Ya., Liubimov, A. N.: The NMR of Polymers (transl.) Plenum Press, New York (1970)Google Scholar
  3. 3.
    Hahn, E. L.: Phys. Rev. 80, 580 (1950)Google Scholar
  4. 4.
    Carr, H. Y., Purcell, E. M.: Phys. Rev. 94, 630 (1954)Google Scholar
  5. 5.
    Stejskal, E. O., Tanner, J. E.: J. Chem. Phys. 42, 288 (1965)Google Scholar
  6. 6.
    Tanner, J. E.: Ph. D. Thesis (Chemistry) University of Wisconsin (1966)Google Scholar
  7. 7.
    Meiboom, S., Gill, D.: Rev. Sci. Instrum. 29, 688 (1958)Google Scholar
  8. 8.
    Tanner, J. E.: J. Chem. Phys. 52, 2523 (1970)Google Scholar
  9. 9.
    Packer, K. J., Rees, C., Tomlinson, D. J.: Molecul. Phys. 18, 421 (1970)Google Scholar
  10. 10.
    Zupančič, I., Pirš, J.: J. Phys. (London) E 9, 79 (1976)Google Scholar
  11. 11.
    Callaghan, P. T., Trotter C. M., Jolley, K. W.: J. Magn. Reson. 37, 247 (1980)Google Scholar
  12. 12.
    Hrovat, M. I., Wade, C. G.: J. Magn. Reson. 44, 62 (1981) and 45, 67 (1981)Google Scholar
  13. 13.
    Cantor D. M., Jonas, J.: J. Magn. Reson. 28, 157 (1977)Google Scholar
  14. 14.
    von Meerwall, E., Burgan, R. D., Ferguson, R. D.: J. Magn. Reson. 34, 339 (1979)Google Scholar
  15. 15.
    Stilbs, P., Moseley, M. E.: Chem. Scripta 15, 176 (1980)Google Scholar
  16. 16.
    Callaghan, P. T., Jolley K. W., Trotter, C. M.: JEOL News 16A, 48 (1980)Google Scholar
  17. 17.
    McCall, D. W., Douglass, D. C., Anderson, E. W.: J. Polym. Sci. A 1, 1709 (1963)Google Scholar
  18. 18.
    McCall, D. W., Huggins, C. M.: Appl. Phys. Letters 7, 153 (1965)Google Scholar
  19. 19.
    von Meerwall, E.: J. Magn. Reson. 50, 409 (1982)Google Scholar
  20. 20.
    Tanner, J. E.: J. Chem. Phys. 69, 1748 (1978)Google Scholar
  21. 21.
    Zientara, G. P., Freed, J. H.: J. Chem. Phys. 72, 1285 (1980)Google Scholar
  22. 22.
    von Meerwall, E., Ferguson, R. D.: J. Chem. Phys. 74, 6956 (1981)Google Scholar
  23. 23.
    von Meerwall, E.: Computer Phys. Commun. 17, 309 (1979)Google Scholar
  24. 24.
    von Meerwall, E., Ferguson, R. D.: Computer Phys. Commun. 21, 421 (1981)Google Scholar
  25. 25.
    Grosz, B., Kosfeld, R.: Messtechnik 7, 171 (1969)Google Scholar
  26. 26.
    deGennes, P. G.: J. Chem. Phys. 55, 572 (1971); also J. Phys. (Paris) 36, 1199 (1975)Google Scholar
  27. 27.
    Doi, M., Edwards, S. F.: J. Chem. Soc. Faraday Trans II 74, 1789, 1802, 1818, (1978); 75, 38 (1979)Google Scholar
  28. 28.
    James, T. L., McDonald, G. G.: J. Magn. Reson. 11, 58 (1973)Google Scholar
  29. 29.
    Boss, B. D., Stejskal, E. O., Ferry, J. D.: J. Phys. Chem. 71, 1501 (1967)Google Scholar
  30. 30.
    von Meerwall, E., Ferguson, R. D.: J. Appl. Polym. Sci. 23, 877 (1979)Google Scholar
  31. 31.
    Ferguson, R. D., von Meerwall, E.: J. Polym. Sci. Polym. Phys. Ed. 18, 1285 (1980)Google Scholar
  32. 32.
    von Meerwall, E., Tomich, D. H., Hadjichristidis, N., Fetters, L. J.: Macromolecules 15, 1157 (1982)Google Scholar
  33. 33.
    Cosgrove, T., Warren, R. F.: Polymer 18, 255 (1977)Google Scholar
  34. 34.
    McCall, D. W., Douglass, D. C., Anderson, E. W.: J. Chem. Phys. 30, 771 (1959)Google Scholar
  35. 35.
    Douglass, D. C. McCall, D. W.: J. Phys. Chem. 62, 1102 (1958)Google Scholar
  36. 36.
    Hirschfelder, J. O., Stevenson, D., Eyring, H.: J. Chem. Phys. 5, 896 (1937)Google Scholar
  37. 37.
    von Meerwall, E., Ferguson, R. D.: J. Chem. Phys. 72, 2861 (1980)Google Scholar
  38. 38.
    McCall, D. W., Anderson, E. W., Huggins, C. M.: J. Chem. Phys. 34, 804 (1961)Google Scholar
  39. 39.
    Tanner, J. E.: Macromolecules 4, 748 (1971)Google Scholar
  40. 40.
    Tanner, J. E., Liu, K.-J., Anderson, J. E.: Macromolecules 4, 586 (1971)Google Scholar
  41. 41.
    von Meerwall, E., Grigsby, J., Tomich, D., Van Antwerp, R.: J. Polym. Sci. Polym. Phys. Ed. 20, 1037 (1982)Google Scholar
  42. 42.
    Bueche, F.: Physical Properties of Polymers, Interscience, New York (1962)Google Scholar
  43. 43.
    Fujita, H.: Fortschr. Hochpolym.-Forsch. 3, 1 (1961)Google Scholar
  44. 44.
    See, for example, Ferry, J. D.: Viscoelastic Properties of Polymers, 3rd ed. Wiley, New York (1980)Google Scholar
  45. 45.
    Flory, P. J.: Principles of Polymer Chemistry, Cornell U. Press, Ithaca, New York (1953), Ch. 14Google Scholar
  46. 46.
    Moseley, M. E.: Polymer Reports 21, 1479 (1980)Google Scholar
  47. 47.
    Pimenov, G. G., Smechko, A. G., Azancheev, N. M., Skurda, V. D.: Akad. Nauk. USSR 20B, 180 (1978)Google Scholar
  48. 48.
    Callaghan P. T., Pinder, D. N.: Macromolecules 13, 1085 (1980)Google Scholar
  49. 49.
    Callaghan, P. T., Pinder, D. N.: Macromolecules 14, 1334 (1981)Google Scholar
  50. 50.
    Pyun, C. W., Fixman, M.: J. Chem. Phys. 41, 937 (1964)Google Scholar
  51. 51.
    Yamakawa, H.: J. Chem. Phys. 36, 2995 (1962)Google Scholar
  52. 52.
    King, T. A., Knox, A., McAdam, J. D. G.: Polymer 14, 293 (1973)Google Scholar
  53. 53.
    von Meerwall, E., Tomich, D. H., Grigsby, J., Pennisi, R., Fetters, L. J., Hadjichristidis, N.: Macromolecules (in press)Google Scholar
  54. 54.
    Callaghan, P. T., Pinder, D. N.: Polymer Bull. 5, 305 (1981)Google Scholar
  55. 55.
    Brown, W., Stilbs, P., Johnsen, R. M.: J. Polym. Sci., Polym. Phys. Ed. 20, 1771 (1982)Google Scholar
  56. 56.
    Kessler, D., Witte, H., Weiss, A.: Ber. Bunsenges, Phys. Chem. 73, 368 (1969)Google Scholar
  57. 57.
    Woessner, D. E.: J. Phys. Chem. 67, 1365 (1963)Google Scholar
  58. 58.
    Kosfeld, R., Goffloo, K.: Kolloid-Z. 247, 801 (1971)Google Scholar
  59. 59.
    Williams, M. L., Landel, R. F., Ferry, J. D.: J. Amer. Chem. Soc. 77, 3701 (1955)Google Scholar
  60. 60.
    Maklakov, A. I., Smechko, A. G., Maklakov, A. A.: Macromolecular Sci. (Moscow) 19, 2611 (1977)Google Scholar
  61. 61.
    Azancheev, N. M., Maklakov, A. I.: Macromolecular Sci. (Moscow) 21, 1574 (1979)Google Scholar
  62. 62.
    Zupančič, I., Lahajnar, G., Blinc, R., Reneker, D. H., Peterlin, A.: J. Polym. Sci., Polym. Phys. Ed. 16, 1399 (1978)Google Scholar
  63. 63.
    Moseley, M. E., Stilbs, P.: Chemica Scripta 16, 114 (1980)Google Scholar
  64. 64.
    Nyström, B., Moseley, M. E., Stilbs, P., Roots, J.: Polymer 22, 218 (1981)Google Scholar
  65. 65.
    von Meerwall, E., Ferguson, R. D.: J. Polym. Sci., Polym. Phys. Ed. 19, 77 (1981)Google Scholar
  66. 66.
    von Meerwall, E., Van Antwerp, R.: Macromolecules 15, 1115 (1982)Google Scholar
  67. 67.
    Vrentas, J. S., Duda, J. L.: J. Polym. Sci., Polym. Phys. Ed. 15, 403 and 417 (1977); 17, 1085 (1979)Google Scholar
  68. 68.
    Derbyshire, W., Duff, I. D.: Discuss Faraday Soc. 57, 243 (1974)Google Scholar
  69. 69.
    Stilbs, P., Lindman, Bj.: J. Magn. Reson. 48, 132 (1982)Google Scholar
  70. 70.
    von Meerwall, E., Ferguson, R. D.: J. Appl. Polymer Sci. 23, 3657 (1979)Google Scholar
  71. 71.
    von Meerwall, E., Ferguson, R. D.: J. Chem. Phys. 75, 937 (1981)Google Scholar
  72. 72.
    Muhr, A. H., Blanchard, J. M. V.: Polymer 23, 1012 (1982) (Suppl.)Google Scholar
  73. 73.
    See, for example, “Diffusion in Polymers”, J. Crank and G. S. Park (eds.) Academic Press, New York (1968)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Ernst D. von Meerwall
    • 1
  1. 1.Physics DepartmentThe University of AkronAkronU.S.A.

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