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Thermal conductivity of a RTV silicone elastomer between 1.2 and 300 K
PII: S0011-2275(97)00146-X
Cryogenics 38 (1998) 227–250 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Thermal conductivity of a RTV silicone elastomer between 1.2 and 300 K A. Baudot, J. Mazuer and J. Odin ` ´ Centre de Recherches sur les Tres Basses Temperatures1/CNRS B.P. 166 38042, Grenoble cedex 09, France
Received 25 July 1997; revised 19 September 1997 The thermal conductivity of a room temperature vulcanized silicone elastomer has been measured between 1.2 and 300 K. The values first increase from 2 × 10-3 W/m.K at 1.2 K to a maximum value of 0.30 W/m.K at 220 K and then quickly drop, remaining at the constant value of 0.27 W/m.K from 240 to 300 K. The maximum corresponds to the melting temperature of the polydimethylsiloxane, the base polymer of the elastomer. A small jump, observed between 75 and 90 K, is attributed to the release of the methyl groups which become free to rotate at upper temperatures. The general behaviour over the whole temperature range is that of the partly crystallized polymers. It is discussed and compared to the behaviour of the completely amorphous materials. 1998 Elsevier Science Ltd. All rights reserved Keywords: polydimethylsiloxane elastomer; thermal conductivity; low temperatures
Introduction Room-temperature vulcanizing (RTV) silicone elastomers are mostly obtained by curing dimethylsiloxane based polymers1. Polydimethylsiloxane is a very unusual polymer having high elasticity even down to low temperatures2. With a glass transition temperature as low as Tg = 150 K, it belongs to the class of polymers with the lowest known Tg values2,3. As a result the silicone elastomers have applications not only in cryogenics as vacuum seals for example but also in various other technological fields4 including medical applications such as artificial heart valves5 or mammary prostheses6. Its remarkable high and low temperature stability is well known and has been used for a long time1,7. Nevertheless to our knowledge, contrary to other elastomers8, no results are available for the low temperature thermal properties of RTV silicones. The aim of this paper is to partly fill this gap by reporting measurements of the thermal conductivity of this material down to 1.2 K. The results compare with those of semi-crystalline polymers9,10 and differ from the thermal conductivity of the widely studied class of fully amorphous materials by the absence of the classical plateau around 10 K. Much interest has been paid to the universal character of the low temperature behaviour of these glasses, plastics and frozen greases11,12 and much theoretical work has been developed to explain the temperature variations of the thermal conductivity over the whole range from very low to ambient temperatures13–17. On the contrary no general analysis is available for the ther1
´ Associated to Universite Joseph Fourier, Grenoble, France.
mal conductivity of semi-crystalline polymer, in spite of a quasi universal behaviour, as we will show.
Experimental procedures We worked with a RTV silicone (Rhodorsil RTV 1556 from Rhone-Poulenc France) available in the uncured state as two components. Following the supplier’s instructions, curing was obtained by mixing the two components at room temperature. Before cross-linking, the mixture was placed under vacuum for about 30 min to eliminate gas bubbles. The material was then ready to be moulded. After degassing we filled a small cylindrical aluminium mould with uncured paste and allowed it to cross-link at room temperature. We obtained a rubber sample 8 mm in diameter and 6 mm in length. This geometry has been chosen to minimize the radiative losses from the lateral area of the sample. The thermal conductivity is measured by using a classical steady state method illustrated in Figure 1, which has been described elsewhere18, and which we summarize below. The sample was glued (with Silastene from Rhone-Poulenc France) onto an isothermal copper block whose temperature is regulated at the desired value To. A small heater also glued with the same paste onto the opposite surface creates a temperature difference ⌬T along the sample placed in an evacuated calorimeter. Calorimeter and sample were first cooled down and the measurements were then performed at increasing temperatures T. The apparent thermal conductivity kapp was deduced from Fourier’s law P = kappg ⌬T, where P is the electrical power supplied to the heater, g = s/l is the geometrical factor of the
Cryogenics 1998 Volume 38, Number 2 227
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al.
Figure 1 Schematic design for the measurement of the thermal conductivity (not to scale).
sample (s, cross section area, l, effective length between temperature sensors). During all the experiments ⌬T was maintained between 1-3% of T. Due to g, the error on the absolute value of kapp was estimated at ⬇ 10%, but the variations of kapp with T were known with a much greater accuracy, of the order of 1%. The reference temperature To of the sample was obtained from calibrated platinum resistors. ⌬T was deduced from the voltage ⌬T of AuFe/chromel P thermocouples that we had previously calibrated19. Both sensors (from Leico Industries, Inc.) were in the form of thin wires of 76.2 m in diameter. Two junctions were made with indium solder and inserted at about 2 mm depth from the lateral surface of the sample into two thin radial slits carefully cut with a needle. From the analysis of the thermoelectric circuit in Figure 1, we deduced: ⌬T = ⌬V/(SAuFe − SK )
(1)
and ⌬To = ⌬Vo/(SAuFe − SK )
Figure 2 The variations with the temperature of -䊊- the apparent (measured) thermal conductivity and -쎲- the actual conductivity (high temperature range only). Also shown for comparison, from 8, the thermal conductivity of a poly(styrenecobutadiene) rubber (solid line) and of a poly(isobutylene-coisoprene) rubber (dashed line).
by using thin wires of about 20 cm in length and is estimated to be around three orders of magnitude less than the heating power. But at temperatures greater than about 100 K, a parasitic radiative heat transfer along the sample cannot be neglected. kapp then incorporates a parasitic radiative term krad which, from the differential form of Stephan’s law, is proportional to T 3. The geometry of the sample is completely determined and at each experimental point the temperatures of the emmitting surfaces (lateral area and bottom area with the heater) are known. So, taking a value of unity for the emissivity of the sample, an upper value for krad(T) can be calculated. In Figure 2 the actual conductivity k = kapp-krad is plotted versus the temperature, together with the measured values kapp. At 90 K, k differs from kapp by only 1% and obviously, when decreasing the temperature, the difference becomes rapidly neglibible and no longer appears in Figure 2.
(2)
Discussion SAuFe and SK are the thermoelectric powers of the couples determined at the mean temperature T of the sample: T = To + ⌬To + ⌬T/2
(3)
Results We show (Figure 2) the variations with temperature of the measured thermal conductivity in the high temperature range. The determination of the actual conductivity k(T) from Fourier’s law as described above is acceptable only when no parasitic heat transfer occurs between the hot source and the cold one. We have avoided convective losses, the calorimeter being under vacuum during the whole experiment. Furthermore, the external surface of the calorimeter is directly in a helium bath at 4.2 K, so that any residual gas is condensed on the cold wall. The possible heat sink through the working thermocouples is minimized
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Cryogenics 1998 Volume 38, Number 2
The thermal properties of the RTV silicone elastomers result from combinations of the properties of the polydimethylsiloxane polymer with the added fillers and of the room temperature vulcanizing process. The fillers (silicates, aluminates, quartz, metal oxides) are added to reinforce the elastomers7. The vulcanizing process determines the chemical nature of the crosslinks and the degree of crosslinking. At ambient temperature the polysiloxane rubbers can be schematically seen as being constituted of a disordered semi-rigid network of crosslinks between which the polydimethylsiloxane chains have the common random coil conformation. The crosslinks network gives to the elastomer its mechanical properties which depends on the fillers and which can be characterized by the tensile strength value. In our case, this value is 7.5 MPa from supplier’s technical brochure. The polydimethylsiloxane chains with their high elasticity form the so-called amorphous part of the elastomer. These originally amorphously oriented chains are known to exhibit unusually high rates of crystallisation7
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al. which increase with decreasing temperature below the melting temperature of the polymer20, Tm = 220 K. In our apparatus, the cooling rate of the sample is of the order of 1 K/min. This rate is slow enough to allow for a large crystallization of the amorphous phase of the elastomer. Here crystallisation must be understood as the conversion of the random coil conformation to a morphology of the elastomer which is not unique and depends on the undercooled state. Various structures generally form this crystalline state which includes regularly coiled helicoidal structures, platelike crystallites and spherullites. Due to the random growth of these crystals, there is a large dispersion of the amorphous domains with a wide distribution of their glass transition temperatures, so that the macroscopic vitreous transition is spread out over a wide temperature range. As a result no remarkable change in k(T) appears at ⬇ 150 K, the glass transition temperature of polydimethylsiloxane. The drop at Tm = 220 K on heating is due to the melting of that part of the elastomer which has crystallized during cooling: the sample then contains more amorphous material having an intrinsically smaller conductivity than the assembly of the coiled helical crystallites. It must be noted that this drop is only 10%. This is a rather small value which is in agreement with the hypothesis of a largely retained coiled helical conformation at low temperatures in the melt2. Another feature which is also characteristic of the polydimethylsiloxane polymer is the rather steep jump observed between 75-90 K. We think that this corresponds to a supplementary contribution to the thermal conductivity due to the releasing of the methyl groups, which become free to rotate7. Over the whole temperature range, the thermal conductivity is mainly determined by the general disordered character of the described morphology, and its values are low, typical of amorphous materials11,14. However, the crystalline part of the elastomer influences the low-temperature behaviour and contrary to the completely amorphous materials, no plateau appears in the k(T) curve around 10 K. As shown in Figure 3, the RTV silicone elastomer exhibit values which are very near to those obtained on
nylon9 and on a 51% crystallinity polyethylene terephtalate10. Above 30 K, except the drop at 220 K and the jump at 75 K, the behaviour does not differ from that of an amorphous material which has been widely analyzed during the last fifteen years13. Strong arguments were recently given16 for the validity of models involving localized vibrational modes over a wide frequency range. The smooth increase with temperature of the thermal conductivity observed at temperatures above 30 K would be due to an additional heat transport mechanism by hopping of localized vibrations, which could be fractons or any other localized vibrational state14. This model assumes the existence of a mobility edge in the phonon spectrum, analogous to the mobility edge for localized electronic states21. This edge is responsible for the thermal conductivity plateau, as did the cutoff frequency in older models11. Vibrational modes with frequencies above the edge are localized and their hopping from site to site is assisted by phonons with frequency below the mobility edge14,15. The thermal conductivity of our sample and of other partly crystalline materials such as natural rubber22 and polymers9,10 is no longer comparable to that of amorphous solids23 between 1-30 K. No plateau is observed and the absolute values of k(T) are lower by one order of magnitude. The width of the plateau might depend upon the strength of the anharmonicity14 and a decrease of the cutoff frequency could explain the absence of the plateau11. From experiments on polyethylene terephtalate samples with adjustable crystallinity, it has been argued10 that the low conductivity could be due to the thermal boundary resistance at crystalline-amorphous interfaces resulting from the acoustic mismatch which introduces a T3 term into the expression of the thermal conductivity24. The influence of crystallinity is important at low temperatures. In polyethylene terephtalate it is evident up to 20 K. The observed k(T) variations are weaker than the theoretical cubic law. Below 10 K, for polymers with a rather high crystallinity of 60 vol.%, a square power law can be roughly considered24. Between 1.2-20 K, our data are fitted by a least square curve of the form k = 0.00214 T1.20 (in W m-1k-1 ) with a mean square deviation less than 0.5‰, not far from the fit for nylon9. At T ⬇ 20 K there is an unexplained change in the slope and between 20 and 75 K, the best fit is obtained with a power law k = 0.0152 T0.56 (mean square deviation less than 1.5%). In this range, crosslink points might act as phonons scattering centers24. Lower temperatures cannot be reached in our apparatus, but results for other elastomers25 are available between 0.1 and 1 K. They show that the slope of k(T) increases and the exponent of the T in the power law reaches 1.8, a value not far from 2. The authors then conclude that in elastomers, as for other amorphous materials at low temperature, the phonons are scattered from localized two level states26.
Conclusion
Figure 3 The variations with the temperature of the -쎲- actual conductivity of the polydimethylsiloxane elastomer together with published results on nylon ( 9 ), polyethylene terephtalate PET- with a 51% volume fraction crystallinity ( 10 ), a-SiO2 and PMMA ( 23 ), natural rubber ( 22 ) and crepe rubber ( 25 ).
The thermal conductivity of the RTV silicon elastomer shows between 1.2 and 300 K a behaviour which is typical of what is observed on crystalline polymers such as nylon. Two sharp changes occur at 75 K and 220 K, which probably correspond respectively to the methyl groups allowed to rotate freely and to the melting temperature of the base polydimethylsiloxane polymer. A change in the slope of
Cryogenics 1998 Volume 38, Number 2 229
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al. k(T) at ⬇ 20 K remained unexplained and does not seem to correspond to any morphological modification of the polymer. At T ⬎ 30 K the k(T) curve is not far from that of completely amorphous polymers and could be interpreted in the frame of model of phonons assisted hopping of localized vibrational states. But for lower temperatures such model predicts the existence of a conductivity plateau, which is absent for partly crystalline amorphous materials. To our knowledge, no general analysis is available for this other type of disordered solids, in spite of an almost universal behaviour, while variations in the low temperature region seem to be mainly associated to the degree of crystallinity.
Acknowledgements We wish to acknowledge M. Grenier for his help during the thermal conductivity measurements and M. Ferrari and P. Brosse-Marron for their technical assistance.Work sup` ported by grants from the Ministere de l’Education nation´ ale, l’Enseignement superieur, la Recherche et l’Insertion ˆ ´ professionnelle and from the Pole Genie Biologique et ´ ´ ˆ Medical de la Region Rhone-Alpes.
References 1. Billmeyer F. W., Jr., Textbook of polymer science. John Wiley and Sons Eds., N.Y., 1971, pp. 482–488. 2. Dvornic P.R. and Lenz R.W., Macromolecules, 1992, 25, 3769. 3. Helmer J.D. and Polmanteer K.E., J. Appl. Polym. Sci., 1969, 13, 2113.
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4. Cornelius D.J. and Monroe C.M., Polym. Eng and Sci., 1985, 25, 467. 5. Parfeev V.M., Grushetskii I.V., Smurova E.V. and Tamuzh V.P., Mech. Compos. Mater., 1981, 17, 297. 6. Piontek R.W. and Kase K.R., Radiology, 1980, 136, 505. 7. Dvornic P.R. and Lenz R.W., High Temperature Siloxane Elasto¨ mers, Huthig and Wepf Verlag: Heidelberg and New York, 1990. 8. Bhowmick T. and Pattanayak S., Cryogenics, 1990, 30, 116. 9. Ashworth T., Loomer J.E. and Kreitman M.M., Adv. Cryog. Eng., 1973, 18, 271. 10. Choy C.L. and Greig D., J. Phys. C: Solid State Phys., 1975, 8, 3121. 11. Anderson A.C., Thermal conductivity in amorphous solids, low temperature properties, Ed. W.A. Phillips, Springer Verlag, Berlin, 1981, p. 65. 12. Zeller R.C. and Pohl R.O., Phys. Rev. B, 1971, 4, 2029. 13. for a brief recall of the main references, see for example Poetsch ¨ R.H.H. and Bottger H., Phys. Rev. B, 1994, 50, 15757. 14. Jagannathan A., Orbach R. and Entin-Wohlman O., Phys. Rev. B, 1989 39, 13465. 15. Michalski J., Phys. Rev. B, 1992, 45, 7054. 16. Orbach R., Physica B, 1996, 219–220, 231. ¨ 17. Bottger H., Damper T. and Poetzsch R., Physica B, 1996, 219– 220, 348. 18. Guessous A. and Mazuer J., J. Physique-Lettres, 1982, 43, L-801. 19. Chaussy J., Guessous A. and Mazuer J., Rev. Sci. Instrum., 1981, 52, 1721. 20. Amaral L.Q., Vinhas L.A. and Herdade S.B., J. Polym. Sci., 1976, 14, 1077. 21. Mott N.F., Philos. Mag., 1969, 19, 835. 22. Freeman J.J. and Greig D., Ad. Cryog. Eng., 1984, 30, 105. 23. Graebner J.E., Golding B. and Allen L.C., Phys. Rev. B, 1986, 34, 5696. 24. Hartwig G., Polymer Properties at Room and Cryogenic Temperatures Plenum Press, New-York, 1994, Chapt. 5. 25. Bhattacharyya A., Smith T.L. and Anderson A.C., J. Non Cryst. Solids, 1979, 31, 395. 26. Freeman J.J. and Anderson A.C., Phys. Rev. B, 1986, 34, 5684.
Cryogenics 38 (1998) 227–250 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Thermal conductivity of a RTV silicone elastomer between 1.2 and 300 K A. Baudot, J. Mazuer and J. Odin ` ´ Centre de Recherches sur les Tres Basses Temperatures1/CNRS B.P. 166 38042, Grenoble cedex 09, France
Received 25 July 1997; revised 19 September 1997 The thermal conductivity of a room temperature vulcanized silicone elastomer has been measured between 1.2 and 300 K. The values first increase from 2 × 10-3 W/m.K at 1.2 K to a maximum value of 0.30 W/m.K at 220 K and then quickly drop, remaining at the constant value of 0.27 W/m.K from 240 to 300 K. The maximum corresponds to the melting temperature of the polydimethylsiloxane, the base polymer of the elastomer. A small jump, observed between 75 and 90 K, is attributed to the release of the methyl groups which become free to rotate at upper temperatures. The general behaviour over the whole temperature range is that of the partly crystallized polymers. It is discussed and compared to the behaviour of the completely amorphous materials. 1998 Elsevier Science Ltd. All rights reserved Keywords: polydimethylsiloxane elastomer; thermal conductivity; low temperatures
Introduction Room-temperature vulcanizing (RTV) silicone elastomers are mostly obtained by curing dimethylsiloxane based polymers1. Polydimethylsiloxane is a very unusual polymer having high elasticity even down to low temperatures2. With a glass transition temperature as low as Tg = 150 K, it belongs to the class of polymers with the lowest known Tg values2,3. As a result the silicone elastomers have applications not only in cryogenics as vacuum seals for example but also in various other technological fields4 including medical applications such as artificial heart valves5 or mammary prostheses6. Its remarkable high and low temperature stability is well known and has been used for a long time1,7. Nevertheless to our knowledge, contrary to other elastomers8, no results are available for the low temperature thermal properties of RTV silicones. The aim of this paper is to partly fill this gap by reporting measurements of the thermal conductivity of this material down to 1.2 K. The results compare with those of semi-crystalline polymers9,10 and differ from the thermal conductivity of the widely studied class of fully amorphous materials by the absence of the classical plateau around 10 K. Much interest has been paid to the universal character of the low temperature behaviour of these glasses, plastics and frozen greases11,12 and much theoretical work has been developed to explain the temperature variations of the thermal conductivity over the whole range from very low to ambient temperatures13–17. On the contrary no general analysis is available for the ther1
´ Associated to Universite Joseph Fourier, Grenoble, France.
mal conductivity of semi-crystalline polymer, in spite of a quasi universal behaviour, as we will show.
Experimental procedures We worked with a RTV silicone (Rhodorsil RTV 1556 from Rhone-Poulenc France) available in the uncured state as two components. Following the supplier’s instructions, curing was obtained by mixing the two components at room temperature. Before cross-linking, the mixture was placed under vacuum for about 30 min to eliminate gas bubbles. The material was then ready to be moulded. After degassing we filled a small cylindrical aluminium mould with uncured paste and allowed it to cross-link at room temperature. We obtained a rubber sample 8 mm in diameter and 6 mm in length. This geometry has been chosen to minimize the radiative losses from the lateral area of the sample. The thermal conductivity is measured by using a classical steady state method illustrated in Figure 1, which has been described elsewhere18, and which we summarize below. The sample was glued (with Silastene from Rhone-Poulenc France) onto an isothermal copper block whose temperature is regulated at the desired value To. A small heater also glued with the same paste onto the opposite surface creates a temperature difference ⌬T along the sample placed in an evacuated calorimeter. Calorimeter and sample were first cooled down and the measurements were then performed at increasing temperatures T. The apparent thermal conductivity kapp was deduced from Fourier’s law P = kappg ⌬T, where P is the electrical power supplied to the heater, g = s/l is the geometrical factor of the
Cryogenics 1998 Volume 38, Number 2 227
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al.
Figure 1 Schematic design for the measurement of the thermal conductivity (not to scale).
sample (s, cross section area, l, effective length between temperature sensors). During all the experiments ⌬T was maintained between 1-3% of T. Due to g, the error on the absolute value of kapp was estimated at ⬇ 10%, but the variations of kapp with T were known with a much greater accuracy, of the order of 1%. The reference temperature To of the sample was obtained from calibrated platinum resistors. ⌬T was deduced from the voltage ⌬T of AuFe/chromel P thermocouples that we had previously calibrated19. Both sensors (from Leico Industries, Inc.) were in the form of thin wires of 76.2 m in diameter. Two junctions were made with indium solder and inserted at about 2 mm depth from the lateral surface of the sample into two thin radial slits carefully cut with a needle. From the analysis of the thermoelectric circuit in Figure 1, we deduced: ⌬T = ⌬V/(SAuFe − SK )
(1)
and ⌬To = ⌬Vo/(SAuFe − SK )
Figure 2 The variations with the temperature of -䊊- the apparent (measured) thermal conductivity and -쎲- the actual conductivity (high temperature range only). Also shown for comparison, from 8, the thermal conductivity of a poly(styrenecobutadiene) rubber (solid line) and of a poly(isobutylene-coisoprene) rubber (dashed line).
by using thin wires of about 20 cm in length and is estimated to be around three orders of magnitude less than the heating power. But at temperatures greater than about 100 K, a parasitic radiative heat transfer along the sample cannot be neglected. kapp then incorporates a parasitic radiative term krad which, from the differential form of Stephan’s law, is proportional to T 3. The geometry of the sample is completely determined and at each experimental point the temperatures of the emmitting surfaces (lateral area and bottom area with the heater) are known. So, taking a value of unity for the emissivity of the sample, an upper value for krad(T) can be calculated. In Figure 2 the actual conductivity k = kapp-krad is plotted versus the temperature, together with the measured values kapp. At 90 K, k differs from kapp by only 1% and obviously, when decreasing the temperature, the difference becomes rapidly neglibible and no longer appears in Figure 2.
(2)
Discussion SAuFe and SK are the thermoelectric powers of the couples determined at the mean temperature T of the sample: T = To + ⌬To + ⌬T/2
(3)
Results We show (Figure 2) the variations with temperature of the measured thermal conductivity in the high temperature range. The determination of the actual conductivity k(T) from Fourier’s law as described above is acceptable only when no parasitic heat transfer occurs between the hot source and the cold one. We have avoided convective losses, the calorimeter being under vacuum during the whole experiment. Furthermore, the external surface of the calorimeter is directly in a helium bath at 4.2 K, so that any residual gas is condensed on the cold wall. The possible heat sink through the working thermocouples is minimized
228
Cryogenics 1998 Volume 38, Number 2
The thermal properties of the RTV silicone elastomers result from combinations of the properties of the polydimethylsiloxane polymer with the added fillers and of the room temperature vulcanizing process. The fillers (silicates, aluminates, quartz, metal oxides) are added to reinforce the elastomers7. The vulcanizing process determines the chemical nature of the crosslinks and the degree of crosslinking. At ambient temperature the polysiloxane rubbers can be schematically seen as being constituted of a disordered semi-rigid network of crosslinks between which the polydimethylsiloxane chains have the common random coil conformation. The crosslinks network gives to the elastomer its mechanical properties which depends on the fillers and which can be characterized by the tensile strength value. In our case, this value is 7.5 MPa from supplier’s technical brochure. The polydimethylsiloxane chains with their high elasticity form the so-called amorphous part of the elastomer. These originally amorphously oriented chains are known to exhibit unusually high rates of crystallisation7
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al. which increase with decreasing temperature below the melting temperature of the polymer20, Tm = 220 K. In our apparatus, the cooling rate of the sample is of the order of 1 K/min. This rate is slow enough to allow for a large crystallization of the amorphous phase of the elastomer. Here crystallisation must be understood as the conversion of the random coil conformation to a morphology of the elastomer which is not unique and depends on the undercooled state. Various structures generally form this crystalline state which includes regularly coiled helicoidal structures, platelike crystallites and spherullites. Due to the random growth of these crystals, there is a large dispersion of the amorphous domains with a wide distribution of their glass transition temperatures, so that the macroscopic vitreous transition is spread out over a wide temperature range. As a result no remarkable change in k(T) appears at ⬇ 150 K, the glass transition temperature of polydimethylsiloxane. The drop at Tm = 220 K on heating is due to the melting of that part of the elastomer which has crystallized during cooling: the sample then contains more amorphous material having an intrinsically smaller conductivity than the assembly of the coiled helical crystallites. It must be noted that this drop is only 10%. This is a rather small value which is in agreement with the hypothesis of a largely retained coiled helical conformation at low temperatures in the melt2. Another feature which is also characteristic of the polydimethylsiloxane polymer is the rather steep jump observed between 75-90 K. We think that this corresponds to a supplementary contribution to the thermal conductivity due to the releasing of the methyl groups, which become free to rotate7. Over the whole temperature range, the thermal conductivity is mainly determined by the general disordered character of the described morphology, and its values are low, typical of amorphous materials11,14. However, the crystalline part of the elastomer influences the low-temperature behaviour and contrary to the completely amorphous materials, no plateau appears in the k(T) curve around 10 K. As shown in Figure 3, the RTV silicone elastomer exhibit values which are very near to those obtained on
nylon9 and on a 51% crystallinity polyethylene terephtalate10. Above 30 K, except the drop at 220 K and the jump at 75 K, the behaviour does not differ from that of an amorphous material which has been widely analyzed during the last fifteen years13. Strong arguments were recently given16 for the validity of models involving localized vibrational modes over a wide frequency range. The smooth increase with temperature of the thermal conductivity observed at temperatures above 30 K would be due to an additional heat transport mechanism by hopping of localized vibrations, which could be fractons or any other localized vibrational state14. This model assumes the existence of a mobility edge in the phonon spectrum, analogous to the mobility edge for localized electronic states21. This edge is responsible for the thermal conductivity plateau, as did the cutoff frequency in older models11. Vibrational modes with frequencies above the edge are localized and their hopping from site to site is assisted by phonons with frequency below the mobility edge14,15. The thermal conductivity of our sample and of other partly crystalline materials such as natural rubber22 and polymers9,10 is no longer comparable to that of amorphous solids23 between 1-30 K. No plateau is observed and the absolute values of k(T) are lower by one order of magnitude. The width of the plateau might depend upon the strength of the anharmonicity14 and a decrease of the cutoff frequency could explain the absence of the plateau11. From experiments on polyethylene terephtalate samples with adjustable crystallinity, it has been argued10 that the low conductivity could be due to the thermal boundary resistance at crystalline-amorphous interfaces resulting from the acoustic mismatch which introduces a T3 term into the expression of the thermal conductivity24. The influence of crystallinity is important at low temperatures. In polyethylene terephtalate it is evident up to 20 K. The observed k(T) variations are weaker than the theoretical cubic law. Below 10 K, for polymers with a rather high crystallinity of 60 vol.%, a square power law can be roughly considered24. Between 1.2-20 K, our data are fitted by a least square curve of the form k = 0.00214 T1.20 (in W m-1k-1 ) with a mean square deviation less than 0.5‰, not far from the fit for nylon9. At T ⬇ 20 K there is an unexplained change in the slope and between 20 and 75 K, the best fit is obtained with a power law k = 0.0152 T0.56 (mean square deviation less than 1.5%). In this range, crosslink points might act as phonons scattering centers24. Lower temperatures cannot be reached in our apparatus, but results for other elastomers25 are available between 0.1 and 1 K. They show that the slope of k(T) increases and the exponent of the T in the power law reaches 1.8, a value not far from 2. The authors then conclude that in elastomers, as for other amorphous materials at low temperature, the phonons are scattered from localized two level states26.
Conclusion
Figure 3 The variations with the temperature of the -쎲- actual conductivity of the polydimethylsiloxane elastomer together with published results on nylon ( 9 ), polyethylene terephtalate PET- with a 51% volume fraction crystallinity ( 10 ), a-SiO2 and PMMA ( 23 ), natural rubber ( 22 ) and crepe rubber ( 25 ).
The thermal conductivity of the RTV silicon elastomer shows between 1.2 and 300 K a behaviour which is typical of what is observed on crystalline polymers such as nylon. Two sharp changes occur at 75 K and 220 K, which probably correspond respectively to the methyl groups allowed to rotate freely and to the melting temperature of the base polydimethylsiloxane polymer. A change in the slope of
Cryogenics 1998 Volume 38, Number 2 229
Thermal conductivity of a RTV silicone elastomer: A. Baudot et al. k(T) at ⬇ 20 K remained unexplained and does not seem to correspond to any morphological modification of the polymer. At T ⬎ 30 K the k(T) curve is not far from that of completely amorphous polymers and could be interpreted in the frame of model of phonons assisted hopping of localized vibrational states. But for lower temperatures such model predicts the existence of a conductivity plateau, which is absent for partly crystalline amorphous materials. To our knowledge, no general analysis is available for this other type of disordered solids, in spite of an almost universal behaviour, while variations in the low temperature region seem to be mainly associated to the degree of crystallinity.
Acknowledgements We wish to acknowledge M. Grenier for his help during the thermal conductivity measurements and M. Ferrari and P. Brosse-Marron for their technical assistance.Work sup` ported by grants from the Ministere de l’Education nation´ ale, l’Enseignement superieur, la Recherche et l’Insertion ˆ ´ professionnelle and from the Pole Genie Biologique et ´ ´ ˆ Medical de la Region Rhone-Alpes.
References 1. Billmeyer F. W., Jr., Textbook of polymer science. John Wiley and Sons Eds., N.Y., 1971, pp. 482–488. 2. Dvornic P.R. and Lenz R.W., Macromolecules, 1992, 25, 3769. 3. Helmer J.D. and Polmanteer K.E., J. Appl. Polym. Sci., 1969, 13, 2113.
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