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Sub- and superdiffusive displacement laws in disordered media probed by NMR techniques

Sub- and superdiffusive displacement laws in disordered media probed by NMR techniques Rainer Kimmich, Yujie Li, German Farrher, Nail Fatkullin, Markus Kehr Sektion Kernresonanzspektroskopie, Universität Ulm, Germany. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000). Outline.

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Sub- and superdiffusive displacement laws in disordered media probed by NMR techniques

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  1. Sub- and superdiffusive displacement laws in disordered media probed by NMR techniques Rainer Kimmich, Yujie Li, German Farrher, Nail Fatkullin, Markus Kehr Sektion Kernresonanzspektroskopie, Universität Ulm, Germany

  2. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

  3. Outline • perspectives of NMR techniques to measure <r2(t)> • over many orders of magnitude of time • Systems showing anomalous transport properties • (fluids in confining geometries, porous media, polymer melts in bulk) • Examples: polymer dynamics, • hydrodynamic dispersion in porous media

  4. (/2)x (/2)x (/2)x Stim. Echo G  time 1 2 1 maximum gradient B0 Magnet wo = 200 MHz Go = 60 T/m z fringe field 9.4 T magnet 89 mm bore NMR diffusometry in the fringe field of a superconducting magnet

  5. Rapid MAGROFI Diffusometry (magnetization grid rotating frame imaging) t diffusion comp. prep. imaging AQ Mz maps after t 6 mm FT 12 mm 4.5 mm 5 mm tp sample 8 mm B. Simon, R. Ki., H. Köstler, J. Magn. Reson. A 118 (1996) 78 B1 gradients (radio frequency field) instead of B0 gradients:

  6. NMR relaxometry due to intermolecular dipolar interactions NMR imaging of interdiffusion of isotopically labeled molecules four decades of time A. Klemm, R. Metzler, R. Ki., Phys. Rev. E 65 (2002) 021112-1 Combination of fringe-field with rotating frame NMR diffusometry (or likewise with the pulsed gradient spin echo (PGSE) variant) water in VitraPor (10-6 m pore size) MAGROFI FFStE

  7. Intermolecular interactions and relative displacements

  8. z, B0 molecular motion molecule “intra” y r “inter” pair of nuclear dipoles x Spin-lattice relaxation by molecular motions homonuclear dipole-dipole coupling dominates for I = 1/2 (e.g. protons) “intra”: reorientations “inter”: relative translations

  9. z‘ l Dr‘(t) r‘(t) r‘(0) y‘ k x‘

  10.  variation of the angular frequency Evaluation of the relative intermolecular mean square displacement from field-cycling NMR relaxometry data • spin-lattice relaxation by dipolar coupling of protons • distinction of intra- and inter-molecular contributions • separable by mixtures of deuterated and undeuterated molecules undeuterated species dilute solution of undeuterated molecules in deuterated matrix

  11. Field-cycling NMR relaxometry B0/T detection 0.5 ... 1.5 polarization relaxation ~s ~ms ~ms 0 t RF t

  12. fringe field NMR diffusometry field-cycling NMR relaxometry

  13. Intramolecular spin-lattice relaxation by chain modes also reflects the mean squared displacement behavior

  14. three different model theories for polymer chain modes •  three different experimental scenarios: • Rouse model • (chain in a viscous medium; no hydrodynamic backflow; • no “entanglements”, i.e. M < Mc) • Renormalized Rouse formalism • (“entanglements”, i.e. M > Mc, t << tterminal) • Tube/reptation concept • (chains confined in nanoscopic “tubes”)

  15. Rouse model: Bead-and-spring chain in a viscous medium without backflow x 0 Solution:Superposition ofdiscrete Rouse relaxation modes with time constants P. E. Rouse, J. Chem. Phys. 21 (1953) 1272 (M<Mc)

  16. relaxation: diffusion: T. N. Khazanovich, Polymer Sci. USSR 4 (1963) 727 N. Fatkullin, R. Ki., H. W. Weber, Phys. Rev. E 47 (1993) 4600 I. Laws for NMR measurands predicted by the Rouse model: (polymer melts; M < Mc ; no “entanglements”)

  17. II. Laws for NMR measurands predicted by the renormalized Rouse formalism: (polymer melts; M > Mc ; “entanglements”; t << tterminal) relaxation: diffusion:

  18. t t d e t s t t R d III. Tube/reptation concept by Doi and Edwards (definition of 4 characteristic time constants)

  19. Laws for NMR measurands predicted by the tube/reptation concept: (polymer melts confined in mesoscopic pores) special evaluation formalism needed! (N. Fatkullin, R. Ki., Phys. Rev. E 52 (1995) 3273) mean squared segment displacement spin-lattice relaxation time limits (I)DE (II)DE (III)DE (IV)DE

  20. Rouse (III)DE (II)DE (I)DE Rouse reptation crossover from “Rouse” to reptation chain dynamics with decreasing tube diameter a) harmonic radial potential theory b) and Monte Carlo simulations of a modified Stockmayer chain model in a tube with hard walls ) A.Denissov, M.Kroutieva, N.Fatkullin, R. Ki., J. Chem. Phys. 116 (2002) 5217

  21. M < Mc , bulk: scen. I • M > Mc , bulk: scen. II • M arb., confined: scen. III  experimental juxtaposition of the three model scenarios

  22. Field-cycling NMR relaxometry at 85°C Rouse bulk PEO 2000, Mw< Mc

  23. Field-cycling NMR relaxometry at 85°C Rouse bulk PEO 2000, Mw<Mc Ren. Rouse bulk PEO 10 000 Mw>Mc

  24. polymer melts confined in pores 100 nm 100 nm 1 mm TEM, replica pore width 10 nm Linear polyethyleneoxide (PEO; Mw=6000) in solid cross-linked polyhydroxyethylmethacrylate (PHEMA) E. Fischer et al., Macromolecules 37 (2004) 3277

  25. Field-cycling NMR relaxometry at 85°C bulk Rouse bulk PEO 2000, Mw<Mc Ren. Rouse bulk PEO 10 000 Mw>Mc PEO 2,000 to 10,000 confined in nanopores from 8 to 60 nm Evaluation of “tube” diameter effective on time scale 10-9 ... 10-5 s: 0.6 nm

  26. Hydrodynamic dispersion

  27. Péclet number simulation of hydrodynamic dispersion

  28. polarization buffer comp. magnet water reservoir 2.4 l HPLC pump porous sample water supply experimental set-up

  29. field gradient pulses 90o 90o 90o 90o 90o echo t Δ τm total displacement time: Δ hydrodynamic dispersion: measurement of incoherent displacements while coherent flow velocity is compensated

  30. water flowing through VitraPor (10-4 m pore size)

  31. Summary • NMR variants promise access to mean squared displacements • in the time range 10-10 s … 100 s • (... and beyond: magnetic resonance imaging of interdiffusion • of isotopically labeled molecules) • hydrodynamic dispersion shows cross-over from sub- • to superdiffusive behavior with increasing Pe • chain dynamics under mesoscopic confinement • reveals characteristic laws of the tube/reptation model

  32. the group in summer … … and in winter recent collaborators: Esteban Anoardo Ioan Ardelean Bogdan Buhai German Farrher Nail Fatkullin Elmar Fischer Ravinath Kausik Markus Kehr Elke Kossel Ravinath Kausik Yujie Li Carlos Mattea … Funding: Deutsche Forschungsgemeinschaft Alexander-von-Humboldt Foundation Volkswagen Foundation

  33. B0 initial final Low-frequency surface relaxation: Reorientation mediated by translational displacements (RMTD) reorientation determined by a) translational diffusion b) surface topology

  34. NMR diffusometry and the tube/reptation concept • anomalous segment diffusion • mean square curvilinear segment displacements dttube diameter (b , N, D0 known)

  35. melts M<Mc “Rouse” PIB, n = 90 MHZ conc. solution

  36. “Renormalized Rouse” spin-lattice relaxation dispersion of polyisobutylene melts Mw>Mc (H.W. Weber, R. K., Macromolecules (26 (1993) 2597)

  37. spin-lattice relaxation dispersion of polyethylene oxide melts (R. K., N. Fatkullin, R.-O. Seitter, K. Gille, J. Chem. Phys. 98 (1998) 2173)

  38. protons: intra- and intersegment interactions deuterons: only intrasegment interactions polyethyleneoxide polybutadiene II II protons protons III III deuterons deuterons R. Ki., N. Fatkullin, R.-O. Seitter, K. Gille, J. Chem. Phys. 98 (1998) 2173

  39. polymers confined in pores melts in bulk (“entangled“ polymers) R. Ki., R. O. Seitter, U. Beginn, M. Möller, N. Fatkullin, Chem. Phys. Letters 307 (1999) 147

  40. NMR 101 102 103 104 105 106 107 108 109 field cycling relaxometry rot. frame relax. conv. relaxometry diffusometry, transverse relaxation, residual spin couplings 10-1 10-4 10-6 10-2 10-3 10-5 10-7 10-8 10-9

  41. mobile linear polyethylene oxide: PEO 2,000: RF = 4 nm PEO 10,000: RF = 9 nm nearest neighbor distance 0.5 nm “tube” effective for diffusion 60 nm “tube” effective for relaxation random coil for Mw=1665 (RF=N1/2b) The corset effect rigid crosslinked HEMA+DMA methacrylate matrix: pore diameters from 8 to 60 nm that is: … up to 122 PEO diameters … up to 15 PEO Flory radii

  42. flow • FC-relaxometry and length scales: • polymer dynamics  the corset effect • surface relaxation mechanisms the flow-relaxation effect Crossover to “bulk” ? Does flow influence T1 ?

  43. Renormalized Rouse formalism Rouse ß Generalized Langevin equation spin-lattice relaxation diffusion

  44. subdiffusive anomalous diffusion: Lévy Brown R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) • “(mutual) obstruction effect”; • Gaussian propagator, D=D(t) • (e.g. single-file diffusion in zeolites, • Rouse mode based diffusion) • “trapping effect”; • non-Gaussian propagator; • waiting time distribution due to “traps” • (e.g. random walk on fractals, reptation) reptation:  “trapping effect”  non-Gaussian propagators  special evaluation theory for spin echo attenuation required!  Elmar Fischer

  45. B0 initial final Low-frequency surface relaxation: Reorientation mediated by translational displacements (RMTD) reorientation determined by a) translational diffusion b) surface topology

  46. superconducting coil superconcuting coil sample and RF coil damping buffers NMR diffusometry in the fringe field of a superconducting magnet 9.4 T, 400 MHz, 10-5 T/m 4.7 T, 200 MHz, 60 T/m

  47. typical echo attenuation curves measured in linear PEO (Mw=11,200) confined in PHEMA pores at 80°C (fringe field technique; 60 T/m; 200 MHz) 1 fitting parameter: pore diameter dpore= (8+/-1) nm echo attenuation formalism: (N. Fatkullin, R. Ki., Phys. Rev. E 52 (1995) 3273)

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