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Low frequency Raman scattering from polystyrene
SpectrochimicaActa. Vol. 34A. pp, 387 to 389 0 Pergamon Press Ltd. 1978. Printed in Great Britain
05%8539/78/040-0387$02.00/0
Low frequency Raman scattering from polystyrene V. N. SANKARANARAYANAN, R. T. BAILEY and A. J. HYDE Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow Gl lXL, Scotland, U.K. (Received 15 Januuq
1977)
Abstract-The Raman spectra of solid polystyrene have been recorded in the region lo-200cm-’ from room temperature down to 29 K. The spectra were reduced using equations derived for disorder-induced scattering. The resulting form of the density of states curves shows a peak at 15 cm-r. This was discussed on the basis of models for the anomalous specific heat of amorphous solids.
INTRODUCTION
The low-temperature thermal properties of amorphous materials is a problem of current interest. The excess specific heat and thermal conductivity of these materials, in comparison with their crystalline modifications, have been attributed to the presence of extra degrees of freedom, the nature of which is still not clear. Careful experiments at very low temperatures show that irrespective of their chemical composition, these disordered systems have specific heats which vary as proportional to T as against T3 for crystalline solids [l]. Theoretical explanations for this behaviour have been attempted in terms of localised two-level systems [2] and localised Einstein oscillators [3]. Recently it has been suggested that the pinning of mobile defects in a solid at low temperatures could give rise to low frequency vibrations which could be responsible for the excess heat capacity of amorphous solids [4]. In this letter we report an investigation of the low frequency Raman spectrum of an amorphous polymer, polystyrene, with a view to obtaining information on the low frequency modes contributing to the anomalous properties. Polystyrene has been shown to exhibit an excess specific heat below about 1 K [S]. No low frequency Raman data are available for amorphous polymers even though many glasses and amorphous semiconductors have been investigated in detail. In the case of amorphous semiconductors, it has been demonstrated that Raman scattering can provide direct information on the amorphous density of states [6,7]. We have also investigated the low frequency Raman spectrum as a function of temperature down to 29 K to determine the temperature dependence of the vibrational density of states.
samples were mounted on the cold tip of a Displex DE202 cryostat, making sure there was good thermal contact between the specimen and the sample holder. The temperature was measured by inserting a Gold (0.07% Fe)-Chrome1 thermocouple into a hole drilled in the sample near the excited volume. It was noticed that the laser beam at 50 mW power raised the sample temperature by 5 degrees. The temperature was directly read on a Thor Cryogenics Kelvin meter and was accurate to + 1 K. Because of the poor thermal conductivity, the radiation losses prevented the temperature of the samples from falling below 29 K, though the base temperature attainable at the cold tip of the Displex itself was below 10 K. The sample temperature was varied using a heater wound around the cold tip. RESULTS
Figure 1 shows the Raman spectra of polystyrene in the region O-200 cm- ’ for various temperatures. The spectrum bears a close resemblance to that of fused
EXPERIMENTAL
The samples used were compression moulded from Shell “crystal grade” atactic polystyrene at 150°C for 10 hr at 140 kg/cm’. The T. of the samnles were 99 + 2°C and the densiry 1.046 + 0.64 g/cc. M, was 847,500 and M, 275,000. The samples used for the Raman study were about 10 x 6 x 3 mm. The spectra were excited using a Coherent Radiation Model 52 Argon laser operating at 5145 A and recorded using a Spex 1301 CompAct Spectrometer. The
I
I Km
I
I 20”
Frequency, cm-’ Fig. 1. Raman spectra of polystyrene in the low frequency region at various temperatures. 387
388
V. N. SANKARANARAYANAN, R. T. BAILEYand A. J. HYDE
silica even though in the latter it extends to about 700 cm- I. As can be seen, there is a broad band at about 90 cm ’ in all the spectra. This band has been studied in detail previously and has been attributed to the torsional motion of the side group [S]. There is another peak at about 20 cn- ’ which becomes increasingly prominent as the temperature is lowered. It is on this feature that attention is focussed in this letter. It is well recognised that the disorder-induced scattering in amorphous materials can modify the low frequency spectrum by affecting the true density of states (DOS). SHUCKER and GAMMON[9] have shown that the Stokes spectrum is given by Cl +
eJ)lsb(4
(1)
where each band of vibrational state b contributes to the spectrum through a coupling constant Ci,@, a/?,$ being determined by the polarizations of the incident and scattered light. The Bose-Einstein factor is n(w), and gb(o) is the density of vibrational states. The frequency dependent factors (l/w) and n(o) change the shape of the contribution from that of a band with density of states gb(co). It is customary to reduce the observed spectrum as If&d
=
Ia,V,y&d
&
=
1
Cbgb(d.
t2)
If the form of the coupling constant is known, then the DOS can be directly obtained from the Raman spectrum. LANNIN [lo] has shown in the case of amorphous semiconductors, that the coupling constant cb varies as o2 for low frequencies. If we assume a similar frequency dependence, we can write for the reduced spectrum,
50
o
294K
.
178K
0
59K
n
29K
100
Frequency,
cm-’
Fig. 2. Reduced Raman spectra of polystyrene at various temperatures, giving the form of the amorphous density of states.
cL7,Yd(4 = I./l,&)
0
_f_ w
1
[n(w) + l] = g(w)
(3)
which gives the “Raman effective” DOS. Figure 2 shows the reduced Raman spectra for various temperatures. If there is no special scattering mechanism effective only at low temperatures, then all the reduced spectra should coincide. This is seen to be the case within experimental error. The room temperature spectrum appears to be deviating considerably from the low temperature spectra which might be due to the slight change in the conditions of illumination and observation caused by the thermal expansion of the cryostat. The DOS curves show two features. A broad shoulder corresponding to the torsional mode which is shifted in frequency due to the reduction process. Also there is a peak at about 15 cm-‘. This latter peak is not clear in the room temperature curve presumably because the Rayleigh line’ is broad and contributes significantly to the intensity of the spectrum. Also this peak does not appear to shift significantly with temperature. The torsion frequency at about 60 cm- 1 also behaves similarly. DISCUSSION
There are several possible origins for the peak in the DOS at 15 cm-‘. In crystalline polymers the low frequency bands have often been attributed to the crystal modes. HENDRA [ 1l] has recently reported the observation of low frequency bands in a number of crystalline polymers and attributes them to the longitudinal acoustic modes or their harmonics. In an amorphous polymer even though lattice phonons in the normal sense cannot propogate, small regions of local order can, in principle, give rise to LA modes. However, with a bulky side group as in polystyrene, this mode should occur at a much lower frequency than in polyethylene chains. ZOLLER et al. [12] have shown that their experimental data on the low temperature specific heat of polystyrene could be fitted by postulating the presence of individual “Einstein oscillators” in addition to the Debye waves. These oscillators were shown to be the acoustic modes corresponding to a linear diatomic chain and it was argued that they should be excitable even at very low temperatures. An order-of-magnitude calculation shows the frequency of these modes to be about 13 cm-‘. The apparent agreement between this value and the peak in the DOS might be fortuitous and should be confirmed by extending the experiments to lower temperatures. A widely accepted model for amorphous solids which explains the low temperature specific heat assumes the existence of two level systems [2]. Tunnelling between these states can occur through thermally activated jumps across a potential barrier. WINTERLING [13] has recently suggested that light might induce transitions between these levels and the frequency shift of the scattered light then would correspond to the energy
Low frequency Raman scattering difference between the two levels, AE. In order that the levels be effective at low temperatures AE should be of the order of kT, k being the Boltzmann constant. For a temperature of 29 K, the frequency shift is calculated to be about 20 cm- ‘. But at 1 K, AE should be less than 1 cm- ’ in order to contribute to the excess specific heat. This means that AE should vary considerabIy with temperature. It is unlikely that such a large change would take place and so it is a reasonable assumption that the energy levels involved in the tunnelling transitions, should be separated by an amount smaller than the frequency shifts investigated in the present investigation. The nature of the molecular motion giving rise to the low frequency peak in the density of states is not clear. We are at the present time extending our studies to include other amorphous polymers which should throw some light on the nature of these modes. Acknowledgements-The authors would like to thank the Science Research Council for substantial grants for the Raman equipment and for the award of a Fellowship to one of them (VNS). They are grateful to D. W. PHILLIPSfor providing samples of polystyrene.
389
from polystyrene REFERENCES
[l] See for example : R. C. ZELLERand R. 0. POHL, Phys. Rev. B4 2029 (1971); R. B. STEPHENS, Pbys. Res. B13, 852 (1976). [2] P. W. ANDERSON,B. I. HALPERINand C. M. VARMA, Phil. Mug. 25, 1 (1972). [3] H. B. ROSENSTOCK,J. Nnn-Cryst. Solids. 7, 123 (1972). [4] P. R. COUCHMAN, C. L. REYNOLDSand R. M. J. CO~~ERILL,Nature 259, 108 (1976). [5] R. B. STEPHENS, G. S. CIELOSZYK and G. L. SALINGER, Phys. Lett. 3SA 215 (1972). [6] M. H. BRODSKY,In: Light Scattering in Solids. (ed. by M. CARDONA)p. 205. Spinger (1975). [7] G. A. N. CONNELL, Phys. Sfat. Solidii (b) 69, 9 (1975). [8] J. J. KIM, J. MCLEISH, A. J. HYDE and R. T. BAILEY, Chem. Phys. Lett. 22,503 (1973). [9] R. SHUKERand R. W. GAMMON,Phys. Rev. Lett. 25, 222 (1970). [lo] J. S. LANNIN,Sokd State Commun. 12,947 (1973). [l l] P. J. HENDRA,In: StructuralStudies ofMacromolecu[es by Spectroscopic Methods, (ed. by K. J. IVIN), p. 973. John Wiley, New York (1976). [12] P. ZOLLER,D. L. FEHL and R. DILLINGER,J. Polymer Sci. (Polymer Phys. Edn.) 11, 1441 (1973). [ 131 G. WINTERLING,Phys. Rev. B12,2432 (1975).
05%8539/78/040-0387$02.00/0
Low frequency Raman scattering from polystyrene V. N. SANKARANARAYANAN, R. T. BAILEY and A. J. HYDE Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow Gl lXL, Scotland, U.K. (Received 15 Januuq
1977)
Abstract-The Raman spectra of solid polystyrene have been recorded in the region lo-200cm-’ from room temperature down to 29 K. The spectra were reduced using equations derived for disorder-induced scattering. The resulting form of the density of states curves shows a peak at 15 cm-r. This was discussed on the basis of models for the anomalous specific heat of amorphous solids.
INTRODUCTION
The low-temperature thermal properties of amorphous materials is a problem of current interest. The excess specific heat and thermal conductivity of these materials, in comparison with their crystalline modifications, have been attributed to the presence of extra degrees of freedom, the nature of which is still not clear. Careful experiments at very low temperatures show that irrespective of their chemical composition, these disordered systems have specific heats which vary as proportional to T as against T3 for crystalline solids [l]. Theoretical explanations for this behaviour have been attempted in terms of localised two-level systems [2] and localised Einstein oscillators [3]. Recently it has been suggested that the pinning of mobile defects in a solid at low temperatures could give rise to low frequency vibrations which could be responsible for the excess heat capacity of amorphous solids [4]. In this letter we report an investigation of the low frequency Raman spectrum of an amorphous polymer, polystyrene, with a view to obtaining information on the low frequency modes contributing to the anomalous properties. Polystyrene has been shown to exhibit an excess specific heat below about 1 K [S]. No low frequency Raman data are available for amorphous polymers even though many glasses and amorphous semiconductors have been investigated in detail. In the case of amorphous semiconductors, it has been demonstrated that Raman scattering can provide direct information on the amorphous density of states [6,7]. We have also investigated the low frequency Raman spectrum as a function of temperature down to 29 K to determine the temperature dependence of the vibrational density of states.
samples were mounted on the cold tip of a Displex DE202 cryostat, making sure there was good thermal contact between the specimen and the sample holder. The temperature was measured by inserting a Gold (0.07% Fe)-Chrome1 thermocouple into a hole drilled in the sample near the excited volume. It was noticed that the laser beam at 50 mW power raised the sample temperature by 5 degrees. The temperature was directly read on a Thor Cryogenics Kelvin meter and was accurate to + 1 K. Because of the poor thermal conductivity, the radiation losses prevented the temperature of the samples from falling below 29 K, though the base temperature attainable at the cold tip of the Displex itself was below 10 K. The sample temperature was varied using a heater wound around the cold tip. RESULTS
Figure 1 shows the Raman spectra of polystyrene in the region O-200 cm- ’ for various temperatures. The spectrum bears a close resemblance to that of fused
EXPERIMENTAL
The samples used were compression moulded from Shell “crystal grade” atactic polystyrene at 150°C for 10 hr at 140 kg/cm’. The T. of the samnles were 99 + 2°C and the densiry 1.046 + 0.64 g/cc. M, was 847,500 and M, 275,000. The samples used for the Raman study were about 10 x 6 x 3 mm. The spectra were excited using a Coherent Radiation Model 52 Argon laser operating at 5145 A and recorded using a Spex 1301 CompAct Spectrometer. The
I
I Km
I
I 20”
Frequency, cm-’ Fig. 1. Raman spectra of polystyrene in the low frequency region at various temperatures. 387
388
V. N. SANKARANARAYANAN, R. T. BAILEYand A. J. HYDE
silica even though in the latter it extends to about 700 cm- I. As can be seen, there is a broad band at about 90 cm ’ in all the spectra. This band has been studied in detail previously and has been attributed to the torsional motion of the side group [S]. There is another peak at about 20 cn- ’ which becomes increasingly prominent as the temperature is lowered. It is on this feature that attention is focussed in this letter. It is well recognised that the disorder-induced scattering in amorphous materials can modify the low frequency spectrum by affecting the true density of states (DOS). SHUCKER and GAMMON[9] have shown that the Stokes spectrum is given by Cl +
eJ)lsb(4
(1)
where each band of vibrational state b contributes to the spectrum through a coupling constant Ci,@, a/?,$ being determined by the polarizations of the incident and scattered light. The Bose-Einstein factor is n(w), and gb(o) is the density of vibrational states. The frequency dependent factors (l/w) and n(o) change the shape of the contribution from that of a band with density of states gb(co). It is customary to reduce the observed spectrum as If&d
=
Ia,V,y&d
&
=
1
Cbgb(d.
t2)
If the form of the coupling constant is known, then the DOS can be directly obtained from the Raman spectrum. LANNIN [lo] has shown in the case of amorphous semiconductors, that the coupling constant cb varies as o2 for low frequencies. If we assume a similar frequency dependence, we can write for the reduced spectrum,
50
o
294K
.
178K
0
59K
n
29K
100
Frequency,
cm-’
Fig. 2. Reduced Raman spectra of polystyrene at various temperatures, giving the form of the amorphous density of states.
cL7,Yd(4 = I./l,&)
0
_f_ w
1
[n(w) + l] = g(w)
(3)
which gives the “Raman effective” DOS. Figure 2 shows the reduced Raman spectra for various temperatures. If there is no special scattering mechanism effective only at low temperatures, then all the reduced spectra should coincide. This is seen to be the case within experimental error. The room temperature spectrum appears to be deviating considerably from the low temperature spectra which might be due to the slight change in the conditions of illumination and observation caused by the thermal expansion of the cryostat. The DOS curves show two features. A broad shoulder corresponding to the torsional mode which is shifted in frequency due to the reduction process. Also there is a peak at about 15 cm-‘. This latter peak is not clear in the room temperature curve presumably because the Rayleigh line’ is broad and contributes significantly to the intensity of the spectrum. Also this peak does not appear to shift significantly with temperature. The torsion frequency at about 60 cm- 1 also behaves similarly. DISCUSSION
There are several possible origins for the peak in the DOS at 15 cm-‘. In crystalline polymers the low frequency bands have often been attributed to the crystal modes. HENDRA [ 1l] has recently reported the observation of low frequency bands in a number of crystalline polymers and attributes them to the longitudinal acoustic modes or their harmonics. In an amorphous polymer even though lattice phonons in the normal sense cannot propogate, small regions of local order can, in principle, give rise to LA modes. However, with a bulky side group as in polystyrene, this mode should occur at a much lower frequency than in polyethylene chains. ZOLLER et al. [12] have shown that their experimental data on the low temperature specific heat of polystyrene could be fitted by postulating the presence of individual “Einstein oscillators” in addition to the Debye waves. These oscillators were shown to be the acoustic modes corresponding to a linear diatomic chain and it was argued that they should be excitable even at very low temperatures. An order-of-magnitude calculation shows the frequency of these modes to be about 13 cm-‘. The apparent agreement between this value and the peak in the DOS might be fortuitous and should be confirmed by extending the experiments to lower temperatures. A widely accepted model for amorphous solids which explains the low temperature specific heat assumes the existence of two level systems [2]. Tunnelling between these states can occur through thermally activated jumps across a potential barrier. WINTERLING [13] has recently suggested that light might induce transitions between these levels and the frequency shift of the scattered light then would correspond to the energy
Low frequency Raman scattering difference between the two levels, AE. In order that the levels be effective at low temperatures AE should be of the order of kT, k being the Boltzmann constant. For a temperature of 29 K, the frequency shift is calculated to be about 20 cm- ‘. But at 1 K, AE should be less than 1 cm- ’ in order to contribute to the excess specific heat. This means that AE should vary considerabIy with temperature. It is unlikely that such a large change would take place and so it is a reasonable assumption that the energy levels involved in the tunnelling transitions, should be separated by an amount smaller than the frequency shifts investigated in the present investigation. The nature of the molecular motion giving rise to the low frequency peak in the density of states is not clear. We are at the present time extending our studies to include other amorphous polymers which should throw some light on the nature of these modes. Acknowledgements-The authors would like to thank the Science Research Council for substantial grants for the Raman equipment and for the award of a Fellowship to one of them (VNS). They are grateful to D. W. PHILLIPSfor providing samples of polystyrene.
389
from polystyrene REFERENCES
[l] See for example : R. C. ZELLERand R. 0. POHL, Phys. Rev. B4 2029 (1971); R. B. STEPHENS, Pbys. Res. B13, 852 (1976). [2] P. W. ANDERSON,B. I. HALPERINand C. M. VARMA, Phil. Mug. 25, 1 (1972). [3] H. B. ROSENSTOCK,J. Nnn-Cryst. Solids. 7, 123 (1972). [4] P. R. COUCHMAN, C. L. REYNOLDSand R. M. J. CO~~ERILL,Nature 259, 108 (1976). [5] R. B. STEPHENS, G. S. CIELOSZYK and G. L. SALINGER, Phys. Lett. 3SA 215 (1972). [6] M. H. BRODSKY,In: Light Scattering in Solids. (ed. by M. CARDONA)p. 205. Spinger (1975). [7] G. A. N. CONNELL, Phys. Sfat. Solidii (b) 69, 9 (1975). [8] J. J. KIM, J. MCLEISH, A. J. HYDE and R. T. BAILEY, Chem. Phys. Lett. 22,503 (1973). [9] R. SHUKERand R. W. GAMMON,Phys. Rev. Lett. 25, 222 (1970). [lo] J. S. LANNIN,Sokd State Commun. 12,947 (1973). [l l] P. J. HENDRA,In: StructuralStudies ofMacromolecu[es by Spectroscopic Methods, (ed. by K. J. IVIN), p. 973. John Wiley, New York (1976). [12] P. ZOLLER,D. L. FEHL and R. DILLINGER,J. Polymer Sci. (Polymer Phys. Edn.) 11, 1441 (1973). [ 131 G. WINTERLING,Phys. Rev. B12,2432 (1975).