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Becher M, Lichtinger A, Minikejew R, Vogel M, Rössler EA. NMR Relaxometry Accessing the Relaxation Spectrum in Molecular Glass Formers. Int J Mol Sci 2022; 23:ijms23095118. [PMID: 35563506 PMCID: PMC9105706 DOI: 10.3390/ijms23095118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2022] [Revised: 04/29/2022] [Accepted: 04/29/2022] [Indexed: 12/10/2022] Open
Abstract
It is a longstanding question whether universality or specificity characterize the molecular dynamics underlying the glass transition of liquids. In particular, there is an ongoing debate to what degree the shape of dynamical susceptibilities is common to various molecular glass formers. Traditionally, results from dielectric spectroscopy and light scattering have dominated the discussion. Here, we show that nuclear magnetic resonance (NMR), primarily field-cycling relaxometry, has evolved into a valuable method, which provides access to both translational and rotational motions, depending on the probe nucleus. A comparison of 1H NMR results indicates that translation is more retarded with respect to rotation for liquids with fully established hydrogen-bond networks; however, the effect is not related to the slow Debye process of, for example, monohydroxy alcohols. As for the reorientation dynamics, the NMR susceptibilities of the structural (α) relaxation usually resemble those of light scattering, while the dielectric spectra of especially polar liquids have a different broadening, likely due to contributions from cross correlations between different molecules. Moreover, NMR relaxometry confirms that the excess wing on the high-frequency flank of the α-process is a generic relaxation feature of liquids approaching the glass transition. However, the relevance of this feature generally differs between various methods, possibly because of their different sensitivities to small-amplitude motions. As a major advantage, NMR is isotope specific; hence, it enables selective studies on a particular molecular entity or a particular component of a liquid mixture. Exploiting these possibilities, we show that the characteristic Cole-Davidson shape of the α-relaxation is retained in various ionic liquids and salt solutions, but the width parameter may differ for the components. In contrast, the low-frequency flank of the α-relaxation can be notably broadened for liquids in nanoscopic confinements. This effect also occurs in liquid mixtures with a prominent dynamical disparity in their components.
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Affiliation(s)
- Manuel Becher
- Nordbayerisches NMR Zentrum, Universität Bayreuth, 95440 Bayreuth, Germany; (M.B.); (A.L.); (R.M.)
| | - Anne Lichtinger
- Nordbayerisches NMR Zentrum, Universität Bayreuth, 95440 Bayreuth, Germany; (M.B.); (A.L.); (R.M.)
| | - Rafael Minikejew
- Nordbayerisches NMR Zentrum, Universität Bayreuth, 95440 Bayreuth, Germany; (M.B.); (A.L.); (R.M.)
| | - Michael Vogel
- Institut für Physik Kondensierter Materie, Technische Universität Darmstadt, 64289 Darmstadt, Germany;
| | - Ernst A. Rössler
- Nordbayerisches NMR Zentrum, Universität Bayreuth, 95440 Bayreuth, Germany; (M.B.); (A.L.); (R.M.)
- Correspondence:
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2
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Nardelli F, Martini F, Carignani E, Rossi E, Borsacchi S, Cettolin M, Susanna A, Arimondi M, Giannini L, Geppi M, Calucci L. Glassy and Polymer Dynamics of Elastomers by 1H-Field-Cycling NMR Relaxometry: Effects of Fillers. J Phys Chem B 2021; 125:4546-4554. [PMID: 33885314 PMCID: PMC8279540 DOI: 10.1021/acs.jpcb.1c00885] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2021] [Revised: 04/07/2021] [Indexed: 11/30/2022]
Abstract
1H spin-lattice relaxation rate (R1) dispersions were acquired by field-cycling (FC) NMR relaxometry between 0.01 and 35 MHz over a wide temperature range on polyisoprene rubber (IR), either unfilled or filled with different amounts of carbon black, silica, or a combination of both, and sulfur cured. By exploiting the frequency-temperature superposition principle and constructing master curves for the total FC NMR susceptibility, χ″(ω) = ωR1(ω), the correlation times for glassy dynamics, τs, were determined. Moreover, the contribution of polymer dynamics, χpol″(ω), to χ″(ω) was singled out by subtracting the contribution of glassy dynamics, χglass″(ω), well represented by the Cole-Davidson spectral density. Glassy dynamics resulted moderately modified by the presence of fillers, τs values determined for the filled rubbers being slightly different from those of the unfilled one. Polymer dynamics was affected by the presence of fillers in the Rouse regime. A change in the frequency dependence of χpol″(ω) at low frequencies was observed for all filled rubbers, more pronounced for those reinforced with silica, which suggests that the presence of the filler particles can affect chain conformations, resulting in a different Rouse mode distribution, and/or interchain interactions modulated by translational motions.
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Affiliation(s)
- Francesca Nardelli
- Dipartimento
di Chimica e Chimica Industriale, Università
di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
| | - Francesca Martini
- Dipartimento
di Chimica e Chimica Industriale, Università
di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
- Centro
per l’Integrazione della Strumentazione Scientifica dell’Università
di Pisa (CISUP), Lungarno
Pacinotti 43, 56126 Pisa, Italy
| | - Elisa Carignani
- Dipartimento
di Chimica e Chimica Industriale, Università
di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
| | - Elena Rossi
- Dipartimento
di Chimica e Chimica Industriale, Università
di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
| | - Silvia Borsacchi
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
- Centro
per l’Integrazione della Strumentazione Scientifica dell’Università
di Pisa (CISUP), Lungarno
Pacinotti 43, 56126 Pisa, Italy
| | | | | | | | - Luca Giannini
- Pirelli
Tyre SpA, Viale Sarca 222, 20126 Milano, Italy
| | - Marco Geppi
- Dipartimento
di Chimica e Chimica Industriale, Università
di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
- Centro
per l’Integrazione della Strumentazione Scientifica dell’Università
di Pisa (CISUP), Lungarno
Pacinotti 43, 56126 Pisa, Italy
| | - Lucia Calucci
- Istituto
di Chimica dei Composti OrganoMetallici, Consiglio Nazionale delle
Ricerche, via G. Moruzzi
1, 56124 Pisa, Italy
- Centro
per l’Integrazione della Strumentazione Scientifica dell’Università
di Pisa (CISUP), Lungarno
Pacinotti 43, 56126 Pisa, Italy
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3
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Flämig M, Hofmann M, Lichtinger A, Rössler EA. Application of proton field-cycling NMR relaxometry for studying translational diffusion in simple liquids and polymer melts. MAGNETIC RESONANCE IN CHEMISTRY : MRC 2019; 57:805-817. [PMID: 30604576 DOI: 10.1002/mrc.4823] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Revised: 12/19/2018] [Accepted: 12/20/2018] [Indexed: 06/09/2023]
Abstract
With the availability of commercial field-cycling relaxometers together with progress of home-built instruments nuclear magnetic resonance relaxometry has gained new momentum as a method of investigating the dynamics in viscous liquids and polymer melts. The method provides the frequency dependence of the spin-lattice relaxation rate. In the case of protons, one distinguishes intramolecular and intermolecular relaxation pathways. Whereas the intramolecular contribution prevails at high frequencies and reflects rotational dynamics, the often ignored intermolecular relaxation contribution dominates at low-frequency and provides access to translational dynamics. A universal low-frequencies dispersion law holds which in pure systems allows determining the diffusion coefficient in a straightforward way. In addition, the rotational time constant is extracted from the high-frequency relaxation contribution. This is demonstrated for simple and ionic liquids and for polymer melts.
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Affiliation(s)
- Max Flämig
- Experimentalphysik and Nordbayerisches NMR-Zentrum, Universität Bayreuth, Bayreuth, Germany
| | - Marius Hofmann
- Experimentalphysik and Nordbayerisches NMR-Zentrum, Universität Bayreuth, Bayreuth, Germany
| | - Anne Lichtinger
- Experimentalphysik and Nordbayerisches NMR-Zentrum, Universität Bayreuth, Bayreuth, Germany
| | - Ernst A Rössler
- Experimentalphysik and Nordbayerisches NMR-Zentrum, Universität Bayreuth, Bayreuth, Germany
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4
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Pestryaev EM. Signature of Reptation in the Long-Time Behavior of the Simulated Free Induction Decay in High Molecular Mass Polymer Melt. POLYMER SCIENCE SERIES A 2019. [DOI: 10.1134/s0965545x19030118] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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5
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Lozovoi A, Mattea C, Fatkullin N, Stapf S. Segmental Dynamics of Entangled Poly(ethylene oxide) Melts: Deviations from the Tube-Reptation Model. Macromolecules 2018. [DOI: 10.1021/acs.macromol.8b01857] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- A. Lozovoi
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
| | - C. Mattea
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
| | - N. Fatkullin
- Institute of Physics, Kazan Federal University, 420008 Kazan, Tatarstan, Russia
| | - S. Stapf
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
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6
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Flämig M, Hofmann M, Rössler EA. Field-cycling NMR relaxometry: the benefit of constructing master curves. Mol Phys 2018. [DOI: 10.1080/00268976.2018.1517906] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- M. Flämig
- Experimentalphysik II, Universität Bayreuth, Bayreuth, Germany
| | - M. Hofmann
- Department of Chemistry, Louisiana State University, Baton Rouge, LA, USA
| | - E. A. Rössler
- Experimentalphysik II, Universität Bayreuth, Bayreuth, Germany
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7
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Mohamed F, Flämig M, Hofmann M, Heymann L, Willner L, Fatkullin N, Aksel N, Rössler EA. Scaling analysis of the viscoelastic response of linear polymers. J Chem Phys 2018; 149:044902. [PMID: 30068172 DOI: 10.1063/1.5038643] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Viscoelastic response in terms of the complex shear modulus G*(ω) of the linear polymers poly(ethylene-alt-propylene), poly(isoprene), and poly(butadiene) is studied for molar masses (M) from 3k up to 1000k and over a wide temperature range starting from the glass transition temperature Tg (174 K-373 K). Master curves G'(ωτα) and G″(ωτα) are constructed for the polymer-specific relaxation. Segmental relaxation occurring close to Tg is independently addressed by single spectra. Altogether, viscoelastic response is effectively studied over 14 decades in frequency. The structural relaxation time τα used for scaling is taken from dielectric spectra. We suggest a derivative method for identifying the different power-law regimes and their exponents along G″(ωτα) ∝ ωε″. The exponent ε″ = ε″(ωτα) ≡ d ln G″(ωτα)/d ln(ωτα) reveals more details compared to conventional analyses and displays high similarity among the polymers. Within a simple scaling model, the original tube-reptation model is extended to include contour length fluctuations (CLFs). The model reproduces all signatures of the quantitative theory by Likhtman and McLeish. The characteristic times and power-law exponents are rediscovered in ε″(ωτα). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (ε″ = -0.25), i.e., G″(ω) ∝ ω-0.21±0.02. At lower frequencies, a second regime with lower exponent ε″ is observed signaling the crossover to coherent reptation. Application of the full Likhtman-McLeish calculation provides a quantitative interpolation of ε″(ωτα) at frequencies below those of the Rouse regime. The derivative method also allows identifying the entanglement time τe. However, as the exponent in the Rouse regime (ωτe > 1) varies along εeRouse = 0.66 ± 0.04 (off the Rouse prediction εRouse = 0.5) and that at ωτe < 1 is similar, only a weak manifestation of the crossover at τe is found at highest M. Yet, calculating τe/τα= (M/Mo)2, we find good agreement among the polymers when discussing ε″(ωτe). The terminal relaxation time τt is directly read off from ε″(ωτα). Plotting τt/τe as a function of Z = M/Me, we find universal behavior as predicted by the TR model. The M dependence crosses over from an exponent significantly larger than 3.0 at intermediate M to an exponent approaching 3.0 at highest M in agreement with previous reports. The frequency of the minimum in G″(ωτα) scales as τmin ∝ M1.0±0.1. An M-independent frequency marks the crossover to glassy relaxation at the highest frequencies. Independent of the amplitude of G″(ω), which may be related to sample-to-sample differences, the derivative method is a versatile tool to provide a detailed phenomenological analysis of the viscoelastic response of complex liquids.
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Affiliation(s)
- F Mohamed
- Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - M Flämig
- Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - M Hofmann
- Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA
| | - L Heymann
- Technische Mechanik und Strömungsmechanik, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - L Willner
- Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - N Fatkullin
- Institute of Physics, Kazan Federal University, Kazan 420008, Tatarstan, Russia
| | - N Aksel
- Technische Mechanik und Strömungsmechanik, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - E A Rössler
- Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany
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8
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Pestryaev EM. Oscillating Free Induction Decay in Polymer Systems: Theoretical Analysis. POLYMER SCIENCE SERIES A 2018. [DOI: 10.1134/s0965545x18040090] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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9
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Yamamoto U, Carrillo JMY, Bocharova V, Sokolov AP, Sumpter BG, Schweizer KS. Theory and Simulation of Attractive Nanoparticle Transport in Polymer Melts. Macromolecules 2018. [DOI: 10.1021/acs.macromol.7b02694] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023]
Affiliation(s)
- Umi Yamamoto
- Department of Physics, University of Illinois, Urbana, Illinois 61801, United States
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States
| | | | | | - Alexei P. Sokolov
- Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, United States
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10
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Kimmich R, Fatkullin N. Self-diffusion studies by intra- and inter-molecular spin-lattice relaxometry using field-cycling: Liquids, plastic crystals, porous media, and polymer segments. PROGRESS IN NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY 2017; 101:18-50. [PMID: 28844220 DOI: 10.1016/j.pnmrs.2017.04.001] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Revised: 04/02/2017] [Accepted: 04/02/2017] [Indexed: 06/07/2023]
Abstract
Field-cycling NMR relaxometry is a well-established technique for probing molecular dynamics in a frequency range from typically a few kHz up to several tens of MHz. For the interpretation of relaxometry data, it is quite often assumed that the spin-lattice relaxation process is of an intra-molecular nature so that rotational fluctuations dominate. However, dipolar interactions as the main type of couplings between protons and other dipolar species without quadrupole moments can imply appreciable inter-molecular contributions. These fluctuate due to translational displacements and to a lesser degree also by rotational reorientations in the short-range limit. The analysis of the inter-molecular proton spin-lattice relaxation rate thus permits one to evaluate self-diffusion variables such as the diffusion coefficient or the mean square displacement on a time scale from nanoseconds to several hundreds of microseconds. Numerous applications to solvents, plastic crystals and polymers will be reviewed. The technique is of particular interest for polymer dynamics since inter-molecular spin-lattice relaxation diffusometry bridges the time scales of quasi-elastic neutron scattering and field-gradient NMR diffusometry. This is just the range where model-specific intra-coil mechanisms are assumed to occur. They are expected to reveal themselves by characteristic power laws for the time-dependence of the mean-square segment displacement. These can be favorably tested on this basis. Results reported in the literature will be compared with theoretical predictions. On the other hand, there is a second way for translational diffusion phenomena to affect the spin-lattice relaxation dispersion. If rotational diffusion of molecules is restricted, translational diffusion properties can be deduced even from molecular reorientation dynamics detected by intra-molecular spin-lattice relaxation. This sort of scenario will be relevant for adsorbates on surfaces or polymer segments under entanglement and chain connectivity constraints. Under such conditions, reorientations will be correlated with translational displacements leading to the so-called RMTD relaxation process (reorientation mediated by translational displacements). Applications to porous glasses, protein solutions, lipid bilayers, and clays will be discussed. Finally, we will address the intriguing fact that the various time limits of the segment mean-square displacement of polymers in some cases perfectly reproduce predictions of the tube/reptation model whereas the reorientation dynamics suggests strongly deviating power laws.
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Affiliation(s)
| | - Nail Fatkullin
- Institute of Physics, Kazan Federal University, Kazan 420008 Tatarstan, Russia
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11
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Lozovoi A, Mattea C, Hofmann M, Saalwaechter K, Fatkullin N, Stapf S. Segmental dynamics of polyethylene-alt-propylene studied by NMR spin echo techniques. J Chem Phys 2017; 146:224901. [PMID: 29166039 DOI: 10.1063/1.4984265] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Segmental dynamics of a highly entangled melt of linear polyethylene-alt-propylene with a molecular weight of 200 kDa was studied with a novel proton nuclear magnetic resonance (NMR) approach based upon 1H → 2H isotope dilution as applied to a solid-echo build-up function ISE(t), which is constructed from the NMR spin echo signals arising from the Hahn echo (HE) and two variations of the solid-echo pulse sequence. The isotope dilution enables the separation of inter- and intramolecular contributions to this function and allows one to extract the segmental mean-squared displacements in the millisecond time range, which is hardly accessible by other experimental methods. The proposed technique in combination with time-temperature superposition yields information about segmental translation in polyethylene-alt-propylene over 6 decades in time from 10-6 s up to 1 s. The time dependence of the mean-squared displacement obtained in this time range clearly shows three regimes of power law with exponents, which are in good agreement with the tube-reptation model predictions for the Rouse model, incoherent reptation and coherent reptation regimes. The results at short times coincide with the fast-field cycling relaxometry and neutron spin echo data, yet, significantly extending the probed time range. Furthermore, the obtained data are verified as well by the use of the dipolar-correlation effect on the Hahn echo, which was developed before by the co-authors. At the same time, the amplitude ratio of the intermolecular part of the proton dynamic dipole-dipole correlation function over the intramolecular part obtained from the experimental data is not in agreement with the predictions of the tube-reptation model for the regimes of incoherent and coherent reptation.
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Affiliation(s)
- A Lozovoi
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
| | - C Mattea
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
| | - M Hofmann
- Department of Chemistry, Louisiana State University, 70803 Baton Rouge, Louisiana, USA
| | - K Saalwaechter
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany
| | - N Fatkullin
- Institute of Physics, Kazan Federal University, Kazan 420008, Tatarstan, Russia
| | - S Stapf
- Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany
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12
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Körber T, Mohamed F, Hofmann M, Lichtinger A, Willner L, Rössler EA. The Nature of Secondary Relaxations: The Case of Poly(ethylene-alt-propylene) Studied by Dielectric and Deuteron NMR Spectroscopy. Macromolecules 2017. [DOI: 10.1021/acs.macromol.6b02536] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Thomas Körber
- Experimentalphysik
II, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Fathia Mohamed
- Experimentalphysik
II, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Marius Hofmann
- Experimentalphysik
II, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Anne Lichtinger
- Experimentalphysik
II, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Lutz Willner
- Institute
of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Ernst A. Rössler
- Experimentalphysik
II, Universität Bayreuth, 95440 Bayreuth, Germany
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13
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Hofmann M, Kresse B, Privalov AF, Heymann L, Willner L, Aksel N, Fatkullin N, Fujara F, Rössler EA. Segmental Mean Square Displacement: Field-Cycling 1H Relaxometry vs Neutron Scattering. Macromolecules 2016. [DOI: 10.1021/acs.macromol.6b01860] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- M. Hofmann
- Experimentalphysik
II, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - B. Kresse
- Institut
für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, D-64289 Darmstadt, Germany
| | - A. F. Privalov
- Institut
für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, D-64289 Darmstadt, Germany
| | - L. Heymann
- Technische
Mechanik und Strömungsmechanik, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - L. Willner
- Institute
of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - N. Aksel
- Technische
Mechanik und Strömungsmechanik, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - N. Fatkullin
- Institute
of Physics, Kazan Federal University, Kazan 420008, Tatarstan Russia
| | - F. Fujara
- Institut
für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, D-64289 Darmstadt, Germany
| | - E. A. Rössler
- Experimentalphysik
II, Universität Bayreuth, D-95440 Bayreuth, Germany
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