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Methyl group rotational tunnelling in glasses: a direct comparison with the crystal
Physica B 276}278 (2000) 361}362
Methyl group rotational tunnelling in glasses: a direct comparison with the crystal A.J. Moreno , A. AlegrmH a , J. Colmenero *, B. Frick Departamento de FnH sica de Materiales, y Centro Mixto CSIC-UPV/EHU, Universidad del PanH s Vasco, Apdo. 1072, 20080 San Sebastia& n, Spain Institute Laue Langevin, BP 156X, F-38042 Grenoble, France
Abstract We present a neutron scattering study of the methyl group dynamics at low temperature in sodium acetate trihydrate, Na(CH COO) ) 3H O, in crystalline and glassy state. As previously found in glassy polymers, the spectra of this low molecular weight system show an apparent quasielastic feature in the glassy state instead of the characteristic rotational tunnelling lines observed in crystals. This can be understood in terms of an asymmetric distribution of rotational tunnelling lines resulting from a Gaussian distribution of potential barriers for methyl rotation. The average barrier of this distribution is found to be notably higher than the unique barrier present in the crystal. 2000 Elsevier Science B.V. All rights reserved. Keywords: Tunnelling; Glasses; Quantum e!ects
1. Introduction Methyl group quantum rotational tunnelling in crystalline systems has been an active "eld of theoretical and experimental investigation over the last 30 years [1]. It is well established that in the low-temperature limit, methyl group dynamics can be explained in terms of a rigid rotator hindered in a single-particle potential, which is required to have the rotational symmetry of the methyl group. Usually, only the threefold term of the Fourier expansion, < (1!cos(3u))/2, is retained. The coupling of the three wells tunnel splits the individual levels and for low barrier systems, the tunnelling frequency lays in the leV range and is re#ected in two inelastic peaks in neutron scattering spectra. However, glassy polymers show an apparent quasielastic feature instead of well-de"ned peaks [2,3]. These results have recently been understood [2,3] in terms of
* Corresponding author. Tel.: #34-943-018205; fax: #34943-212236. E-mail address: [email protected] (J. Colmenero)
a simple model } rotation rate distribution model (RRDM) } which assumes a Gaussian distribution of threefold potential barriers g(< ) for methyl group re orientation. The underlying physical idea is that, in a glassy system, the disorder is re#ected in a di!erent environment for each methyl group, and consequently, in a di!erent barrier. The direct relationship between the height of the barrier, < , and the tunnelling frequency, u , allows to obtain the distribution of tunnelling frequencies in a straightforward way: h( u )"!g(< ) d< /d u . (1) Due to the mainly exponential dependence of u on < , [3] the distribution of tunnelling frequencies is strongly asymmetric [2,3]. Due to this distribution and the "nite energy resolution of the neutron spectrometers, the tunnelling spectrum is observed as `quasielastica, though being inelastic. This model has been successfully applied to glassy polymers and a similar behaviour should be expected in low molecular weight glasses. Such systems have the advantage, that, in contrast to glassy polymers, it is possible to make a direct comparison in the same sample in crystalline and glassy state. The selected system was sodium acetate trihydrate, Na(CH COO) . 3H O,
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 5 6 3 - X
362
A.J. Moreno et al. / Physica B 276}278 (2000) 361}362
which can be considered as a canonical system for methyl group rotational tunnelling in the crystalline state [4,5].
S (Q, u) is the incoherent scattering function of the 00"+ RRDM [2,3]:
2. Experimental
S (Q, u)" d( u )h( u )S (Q, u, u ), 00"+ +%
(3)
where Inelastic measurements were carried out in the backscattering spectrometer IN16 of the ILL (Grenoble, France). A #at sample of thickness 0.20 mm was used to get a transmission higher than 90%, allowing to neglect multiple scattering e!ects. In order to obtain the glassy state of the sample, the temperature was raised above the melting point (803C) but controlled to remain below the boiling point of water, to avoid dehydration. After the melting, a quick cooling down to 1.4 K was applied. Once the measurements in the glassy state were carried out, the sample was crystallised in situ and measured again.
3. Results and discussion Two inelastic lines at $5.8 leV were obtained in the crystalline sample, in complete agreement with the previous measurements by Clough et al. [5]. Fig. 1 shows the spectrum of the glassy sample versus the crystal. As previously seen in glassy polymers, an apparent quasielastic feature is present instead of the two tunnelling peaks. The spectra of the glassy sample were "tted to R(Q, u) S(Q, u), with R(Q, u) the resolution function and S(Q, u)"(p#p !p )d(u) +% #p S (Q, u) (2) +% 00"+ being p, p the total coherent and incoherent cross sections of the sample and p the incoherent cross +% section of the three protons in the methyl group.
5#4 j (Qr) d(u), S (Q, u, u )" +% 9 2(1!j (Qr)) # [d(u!u )#d(u#u )], 9 (4) is the well-known incoherent scattering function for quantum rotational tunnelling in a threefold potential [1], j being the zeroth-order Bessel function and r the H}H distance in the methyl group. The parameters of the Gaussian distribution of potential barriers are directly obtained from Eq. (1). The main results obtained are: (i) the average potential barrier in the glassy state is notably higher than the unique barrier in the crystal (592 versus 403 K); (ii) This average barrier is similar to that of a glassy polymer, PVAc, which has the same ester group [2,6]; (iii) The half-width of the distribution of potential barriers, 218 K, is similar to those found in glassy polymers [6]. These results con"rm the idea that the disorder e!ects on the tunnelling CH -spectra are similar in any glassy state and they should be controlled by the non-bonded interactions.
Acknowledgements We acknowledge the support from the projects: DGICYT, PB97-0638; GV, EX 1998-23 and PI-1998-20; UPV/EHU, 206.215-G20/98. A.J.M acknowledges a grant of the Basque Government.
References
Fig. 1. IN16 spectrum at 1.4 K and Q"1.8 As \. Triangles and circles correspond, respectively, to the sample in crystalline and glassy state. The dashed line is the "t to the RRDM. Thick solid lines are the convoluted distributions of tunnelling lines.
[1] M. Prager, A. Heidemann, Chem. Rev. 97 (1997) 2933. [2] J. Colmenero, R. Mukhopadhyay, A. AlegrmH a, B. Frick, Phys. Rev. Lett. 80 (1998) 2350. [3] A.J. Moreno, A. AlegrmH a, J. Colmenero, B. Frick, Phys. Rev. B 59 (1999) 5983. [4] S. Clough, A. Heidemann, M.N.J. Paley, J.B. Suck, J. Phys. C 13 (1980) 6599. [5] S. Clough, A. Heidemann, M.N.J. Paley, J. Phys. C 14 (1981) 1001. [6] R. Mukhopadhyay, A. AlegrmH a, J. Colmenero, B. Frick, Macromolecules 31 (1998) 3985.
Methyl group rotational tunnelling in glasses: a direct comparison with the crystal A.J. Moreno , A. AlegrmH a , J. Colmenero *, B. Frick Departamento de FnH sica de Materiales, y Centro Mixto CSIC-UPV/EHU, Universidad del PanH s Vasco, Apdo. 1072, 20080 San Sebastia& n, Spain Institute Laue Langevin, BP 156X, F-38042 Grenoble, France
Abstract We present a neutron scattering study of the methyl group dynamics at low temperature in sodium acetate trihydrate, Na(CH COO) ) 3H O, in crystalline and glassy state. As previously found in glassy polymers, the spectra of this low molecular weight system show an apparent quasielastic feature in the glassy state instead of the characteristic rotational tunnelling lines observed in crystals. This can be understood in terms of an asymmetric distribution of rotational tunnelling lines resulting from a Gaussian distribution of potential barriers for methyl rotation. The average barrier of this distribution is found to be notably higher than the unique barrier present in the crystal. 2000 Elsevier Science B.V. All rights reserved. Keywords: Tunnelling; Glasses; Quantum e!ects
1. Introduction Methyl group quantum rotational tunnelling in crystalline systems has been an active "eld of theoretical and experimental investigation over the last 30 years [1]. It is well established that in the low-temperature limit, methyl group dynamics can be explained in terms of a rigid rotator hindered in a single-particle potential, which is required to have the rotational symmetry of the methyl group. Usually, only the threefold term of the Fourier expansion, < (1!cos(3u))/2, is retained. The coupling of the three wells tunnel splits the individual levels and for low barrier systems, the tunnelling frequency lays in the leV range and is re#ected in two inelastic peaks in neutron scattering spectra. However, glassy polymers show an apparent quasielastic feature instead of well-de"ned peaks [2,3]. These results have recently been understood [2,3] in terms of
* Corresponding author. Tel.: #34-943-018205; fax: #34943-212236. E-mail address: [email protected] (J. Colmenero)
a simple model } rotation rate distribution model (RRDM) } which assumes a Gaussian distribution of threefold potential barriers g(< ) for methyl group re orientation. The underlying physical idea is that, in a glassy system, the disorder is re#ected in a di!erent environment for each methyl group, and consequently, in a di!erent barrier. The direct relationship between the height of the barrier, < , and the tunnelling frequency, u , allows to obtain the distribution of tunnelling frequencies in a straightforward way: h( u )"!g(< ) d< /d u . (1) Due to the mainly exponential dependence of u on < , [3] the distribution of tunnelling frequencies is strongly asymmetric [2,3]. Due to this distribution and the "nite energy resolution of the neutron spectrometers, the tunnelling spectrum is observed as `quasielastica, though being inelastic. This model has been successfully applied to glassy polymers and a similar behaviour should be expected in low molecular weight glasses. Such systems have the advantage, that, in contrast to glassy polymers, it is possible to make a direct comparison in the same sample in crystalline and glassy state. The selected system was sodium acetate trihydrate, Na(CH COO) . 3H O,
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 5 6 3 - X
362
A.J. Moreno et al. / Physica B 276}278 (2000) 361}362
which can be considered as a canonical system for methyl group rotational tunnelling in the crystalline state [4,5].
S (Q, u) is the incoherent scattering function of the 00"+ RRDM [2,3]:
2. Experimental
S (Q, u)" d( u )h( u )S (Q, u, u ), 00"+ +%
(3)
where Inelastic measurements were carried out in the backscattering spectrometer IN16 of the ILL (Grenoble, France). A #at sample of thickness 0.20 mm was used to get a transmission higher than 90%, allowing to neglect multiple scattering e!ects. In order to obtain the glassy state of the sample, the temperature was raised above the melting point (803C) but controlled to remain below the boiling point of water, to avoid dehydration. After the melting, a quick cooling down to 1.4 K was applied. Once the measurements in the glassy state were carried out, the sample was crystallised in situ and measured again.
3. Results and discussion Two inelastic lines at $5.8 leV were obtained in the crystalline sample, in complete agreement with the previous measurements by Clough et al. [5]. Fig. 1 shows the spectrum of the glassy sample versus the crystal. As previously seen in glassy polymers, an apparent quasielastic feature is present instead of the two tunnelling peaks. The spectra of the glassy sample were "tted to R(Q, u) S(Q, u), with R(Q, u) the resolution function and S(Q, u)"(p#p !p )d(u) +% #p S (Q, u) (2) +% 00"+ being p, p the total coherent and incoherent cross sections of the sample and p the incoherent cross +% section of the three protons in the methyl group.
5#4 j (Qr) d(u), S (Q, u, u )" +% 9 2(1!j (Qr)) # [d(u!u )#d(u#u )], 9 (4) is the well-known incoherent scattering function for quantum rotational tunnelling in a threefold potential [1], j being the zeroth-order Bessel function and r the H}H distance in the methyl group. The parameters of the Gaussian distribution of potential barriers are directly obtained from Eq. (1). The main results obtained are: (i) the average potential barrier in the glassy state is notably higher than the unique barrier in the crystal (592 versus 403 K); (ii) This average barrier is similar to that of a glassy polymer, PVAc, which has the same ester group [2,6]; (iii) The half-width of the distribution of potential barriers, 218 K, is similar to those found in glassy polymers [6]. These results con"rm the idea that the disorder e!ects on the tunnelling CH -spectra are similar in any glassy state and they should be controlled by the non-bonded interactions.
Acknowledgements We acknowledge the support from the projects: DGICYT, PB97-0638; GV, EX 1998-23 and PI-1998-20; UPV/EHU, 206.215-G20/98. A.J.M acknowledges a grant of the Basque Government.
References
Fig. 1. IN16 spectrum at 1.4 K and Q"1.8 As \. Triangles and circles correspond, respectively, to the sample in crystalline and glassy state. The dashed line is the "t to the RRDM. Thick solid lines are the convoluted distributions of tunnelling lines.
[1] M. Prager, A. Heidemann, Chem. Rev. 97 (1997) 2933. [2] J. Colmenero, R. Mukhopadhyay, A. AlegrmH a, B. Frick, Phys. Rev. Lett. 80 (1998) 2350. [3] A.J. Moreno, A. AlegrmH a, J. Colmenero, B. Frick, Phys. Rev. B 59 (1999) 5983. [4] S. Clough, A. Heidemann, M.N.J. Paley, J.B. Suck, J. Phys. C 13 (1980) 6599. [5] S. Clough, A. Heidemann, M.N.J. Paley, J. Phys. C 14 (1981) 1001. [6] R. Mukhopadhyay, A. AlegrmH a, J. Colmenero, B. Frick, Macromolecules 31 (1998) 3985.