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Thermal conductivity of some silicate glasses in relation to composition and structure
Journal of Non-Crystalline Solids 53 (1982) 165-172 North-Holland Publishing Company
165
T H E R M A L C O N D U C T I V I T Y O F S O M E S I L I C A T E G L A S S E S IN RELATION TO COMPOSITION AND STRUCTURE
M.M. AMMAR, S. G H A R I B and M.M. H A L A W A National Institute For Standards, Dokki, Cairo, Egypt
Kh. E1 B A D R Y , N.A. G H O N E I M and H.A. E1 B A T A L Glass Research Laboratory, National Research Centre, DokkL Cairo, Egypt
Received 15 October 1981 Revised manuscript received 18 May 1982
The thermal conductivities of some silicate glasses have been investigated by a steady-state method at 30°C. Experimental results show that the thermal conductivity progressively decreases with the increase of the formula weight of the R20 component. The effects of composition and structure on thermal conductivity are discussed.
1. Introduction M a n y of the practical applications of glasses in glass solders and glass coatings require accurate knowledge of the thermal conductivity. Heat transfer properties are also of great importance in glass melting, annealing and forming. Several authors [1-8] assumed that this property could be considered as a structure-sensitive property that responds strongly to variation in the chemical composition of the glass. A similar contribution to thermal conductivity of glass was shown by titania [5,7], lime [8] or lead oxide [9]. It was assumed that the thermal conductivity decreased when the composition of the glass was made more complex as a result of the shortening of p h o n o n mean free path because of the increased disorder of the glass structure.
2. Experimental procedure 2.1. Glass p r e p a r a t i o n
The glass compositions are reported in tables 1 and 2. The batches were prepared from Analar and reagent grade carbonates of sodium, calcium, 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
166
M.M. Ammar et al. / Thermal conductivity of silicate glasses
strontium, barium, magnesium and zinc oxide. Silica was introduced as pulverized and purified quartz of high purity. The batches were melted in platinum crucibles using an electric furnace with SiC heaters. The melting temperature was 1400°C. To ensure homogenization the melt was stirred from time to time. The melt was cast into discs which were annealed, ground and polished to a smooth flat parallel surface. The discs were between 18 m m and 5 m m in diameter and 5 m m height.
2.2. Thermal conductivity measurements Thermal conductivity measurements were carried out using an apparatus which has been described in previous articles [5,8]. In this method heat from a heat source flows through the sample under test to a heat sink which transfers this heat by thermal radiation to a surrounding evacuated chamber which has the internal surface blackened. The temperature TO of the heat source is kept constant at T t > TO during the experiment. The emissivity C of the heat sink was measured by using a standard glass sample of known thermal conductivity, and was found to be 0.94. After thermal equilibrium is reached the temperatures of the heat source, the heat sink, and the outer chamber are measured by the attached thermocouple using a Pye precision decade potentiometer to measure the thermal e.m.f, with 0.2 ~tV precision, i.e. the temperature can be measured to within 0.005°C.
2.3. Density and refractive index measurements The density of the glass samples under investigation was measured by the Archimedes method using xylene at 25°C as the immersion liquid. The refractive index was measured with an Abbe refractometer. A thin film of monobromonaphthalene liquid was used as a contact layer between the prism of the refractometer and the test glass sample.
3. Results
The data obtained are listed in tables 1 and 2. Each value of thermal conductivity is the average of at least six determinations. The uncertainty is about + 2%. From table 1 it is seen that there is a change in the thermal conductivity with a change in the content of any of the divalent metal oxides MgO, CaO, SrO, or BaO in the base soda silica glass. The glass containing MgO has the highest thermal conductivity and that containing BaO has the lowest. This result agrees with the findings of previous authors [2,3,5,8] that the oxides with the lowest molecular weight, will generally have the highest contribution to the thermal conductivity. The density and refractive index of the glass increases with the addition of BaO more than SrO and the latter more than CaO.
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From table 2 it is shown that the progressive addition of ZnO to the lithium oxide glass increases the thermal conductivity, density and refractive index of the glass.
4. Discussion 4.1. Theoretical considerations
In a nonmetal heat is conducted by means of the thermal vibration of the atoms [11]. In a simple metal this mode of heat transport makes some contribution, but the observed thermal conductivity is almost entirely due to the electrons. It does not follow that the thermal conductivities of the two types of solid must be very different in magnitude, as they are in the case of electrical conductivity, but their dependence on temperature and on the imperfections in individual specimens are rather different. The process of thermal energy transfer is considered to be a random process [ 12]. The energy does not simply enter one end of the specimen and proceeds directly in a straight path to the other end, but the energy diffuses through the specimen, suffering frequent collisions. Heat is considered as being transmitted by phonons, which are the quanta of energy in each mode of vibration, and the mean free path is a measure of the rate at which energy is exchanged between different phonon modes. In terms of phonons, a flow of heat implies that the phonon distribution differs from that in thermal equilibrium, which corresponds to no net heat flow. The conductivity depends on the extent to which the distribution can depart from equilibrium for a given temperature gradient. The departure from equilibrium, which is expressed in terms of relaxation times or mean free paths, depends, in general, on the temperature and on the frequency and polarization of the phonon mode. It is the random nature of the conductivity process that brings the temperature gradient into the expression for the thermal flux. The thermal conductivity values of several noncrystalline solids are comparable and assumed to be practically identical [ 11-13]. Some authors [ 13,14] believed that there is a complete lack of sensitivity to composition in the noncrystalline solids. This would be understandable if the thermal conductivity values were as low as they could possibly be, corresponding to a diffusion of the vibration energy from one atom to the neighbouring ones. But this is not the case. The phonon mean free path is determined principally by two processes [12], geometrical scattering and scattering by other phonons. If the forces between atoms are purely harmonic, there will be no mechanism for collisions between different phonons, and the mean free path will be limited solely by collisions of a phonon with the crystal boundary and by lattice imperfections. With anharmonic lattice interactions there is a coupling between different phonons which limits the value of the mean free path.
170
M.M. A mmaret al. / Thermal conductivity of silicate glasses
The mean free path in silica glass at room temperature is 8 ,~, which is of the order of magnitude of the dimensions of a silicon dioxide tetrahedron SiO4 (7 ,~) [11-12]. The accepted concept of the nature of the glassy state [15-17] pictures silica glass as an example of a random, but continuous, network of silicon-oxygen bonds. The effective glass network size is only of the order of a single tetrahedron of the stucture. It is expected, therefore (except at very low temperatures where the phonon wavelengths are long), that the phonon mean free path will be constant, limited by the essential structural building unit size. The decline in the thermal conductivity as the temperature is lowered may thus be attributed to the decline in the heat capacity. Two models have been suggested to explain the thermal conductivity of glasses. In the first model [18], Klemens proposed that the scattering is caused by spatial fluctuations in the sound velocity and by a certain amount of long-range order and that this affects the transverse phonons more than the longitudinal ones. At low temperatures the heat is mainly carried by longitudinal phonons. As an alternative model, it is further proposed [19] that in noncrystalline solids the heat-carrying plane-wave phonons are resonantly scattered by localized phonons in a way similar to the resonant scattering found in crystals containing substantial molecules with low-frequency quasirotational degrees of freedom or in crystals containing defects causing resonant modes [20]. These two modes suggest that the scattering must have a very simple origin, quite independent of structural details or the vibrational spectrum of the solids. At high temperatures, the mean free paths for crystalline and vitreous materials approach the order of a few angstroms, i.e. the interatomic separation, as is to be expected. From the previous considerations, it seems logical to suggest that the thermal conductivity must decrease with increase of the disordering of the glass network structure and this results in a shortening of the phonon free path. The same conclusion has been arrived at by Muratov and Chernyshov [6]. To understand and interpret the effect on the thermal conductivity values of adding various monovalent or divalent oxides, the effects of such oxides on the geometrical arrangement of the building units of the glassy network are considered.
4.2. Interpretation of the results When an alkali or alkaline earth oxide reacts with silica to form a glass, the silicon-oxygen network is broken up by the alkali or alkaline earth ions as evidenced by the much lower viscosity of these glasses compared with fused silica [21,22]. So long as the number of A 2 0 or AO units, A being the metallic ion, is in less than a one-to-one ratio to the number of SiO 2 units, the silicon-oxygen tetrahedron is linked to at least three other tetrahedra, and the glass-forming tendency of the mixture is retained. As more alkali or alkaline earth oxide is added beyond the one-to-two ratio, the network becomes more and more disrupted, as more tetrahedra are bonded
M.M. A mmaret al. / Thermal conductivity of silicate glasses
171
to only two other tetrahedra. Glass formation becomes progressively more difficult because the rates of nucleation and crystallization in the glass become much more rapid as alkali or alkaline earth oxide is added. In the composition range between the disilicate (A 20, 2SiO 2 or A0.2 SiO 2) and metasilicate (A20, SiO 2 or A 0 ~. SiO2) chain-like structures should exist, although direct evidence of them is lacking. At alkali concentrations above the metasilicate isolated islands and rings; as well as chains of silicon-oxygen tetrahedra, should be the major structural features. Qualitatively it seems probable that the alkali ion not only causes a loosening of the network, but also introduces weak bonds [23]. With respect to the divalent oxides, the results can be explained by considering the type and quantity of the introduced cations on the same basis as considered for other physical properties [24,25]. These divalent cations can be present in glasses as network-modifiers within interstices or in some special oxides, as network-building tetrahedra groups. Various investigators reported that the zinc cation could have an oxygen ligancy of four or six, or both in glasses [26-30]. The substitution of ZnO for Li20 on a mole basis keeps the number of silicons and oxygens constant but reduces the number of cations per anion, allowing greater freedom for the transverse oxygen vibration [30]. Our data suggest that the zinc cations in such glass compositions show marked preference for the tetrahedral site. This is in agreement with previous assumptions [30].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16 [17] [18] [19]
G,W. Morey, The Properties of Glass (Reinhold, New York, 1954) pp. 217-20. E.H. Ratcliffe, Glass Technol. 4 (1963) 113. P.F. Van Velden, Glass Technol. 6 (1965) 166. D.G. Holloway, The Physical Properties of Glasses (Wykeham, London, 1973) pp. 41-48. M.M. Ammar, Kh. El Badry, M.R. Moussa, M.M. Halawa and S. Gharib, Cent. Glass and Ceram. Res. Inst. Bull. 22 (1975) 10. Muratov and A.V. Chernyshov, Sov. J. Glass Phys. Chem. 5 (Engl. Transl., Consultants Bureau, New York, 1979) p. 100. N.A. Ghoneim, Sprechsaal, 113 (1980) 610. N.A. Ghoneim, A.F. Abbas, M.M. Ammar and M.M. Halawa, Sprechsaal 114 (1981) 293. H.A. E1 Batal et al. Cent. Glass and Ceram. Res. Inst. Bull. (in press). G. Haacke and D.P. Spitzer, J. Sci. Instrum. 42 (1965) 702. R. Berman, Thermal Conduction In Solids (Clarendon Press, Oxford, 1976). C. Kittel, Introduction to Solid State Physics, 2nd Ed. (Wiley, London, 1962) Chap. 6, p. 138. R. Berman, Phys. Rev. 76 (1949) 315. R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. W.H. Zachariasen, J. Amer. Chem. Soc. 54 (1932) 3841. J. Biscoe and B.E. Warren, J. Amer. Ceram. Soc. 21 (1938) 287. R.H. Doremus, Glass Science (Wiley, New York, 1973) pp. 23-43. P.G. Klemens, Proc. R. Soc. A208 (1951) 108. P.G. Klemens, in: Non-crystalline Solids, Ed. J.A. Prins (North-Holland, Amsterdam, 1965). p. 162.
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M.M. A mmar et al. / Thermal conductivity of silicate glasses
[20] R.O. Pohl, in: Localized Excitations in Solids, Ed. R.F. Wallis (Plenum Press, New York, 1968) p. 324. [21] R.H. Doremus, Ann. Rev. Mat. Sci. 2 (1972) 93. [22] R.H. Doremus, Glass Science (Wiley, New York, 1973) pp. 23-31. [23] J.E. Stanworth, Physical Properties of Glass (Oxford University Press, Oxford, 1950). [24] H.A. El Batal et al. Ceram. Bull. Amer. Ceram. Soc. (1982) (in press). [25] Kh. E1 Badry et al. Sprechsaal (1982) (in press). [26] E.M. Levin and S. Block, J. Amer. Ceram. Soc. 40 (1957) 95. [27] A.G.F. Dingwall and H. Moore, J. Soc. Glass Technol. 37 (1953) 316. [28] C. Hirayama, J. Amer. Ceram. Soc. 44 (1961) 602. [29] A.R. Khan and H.E. Simpen, Glass Ind. 31 (1950) 407, 408, 428. [30] J.C. Hurt and C.J. Philips, J. Amer. Ceram. Soc. 53 (1970) 269.
165
T H E R M A L C O N D U C T I V I T Y O F S O M E S I L I C A T E G L A S S E S IN RELATION TO COMPOSITION AND STRUCTURE
M.M. AMMAR, S. G H A R I B and M.M. H A L A W A National Institute For Standards, Dokki, Cairo, Egypt
Kh. E1 B A D R Y , N.A. G H O N E I M and H.A. E1 B A T A L Glass Research Laboratory, National Research Centre, DokkL Cairo, Egypt
Received 15 October 1981 Revised manuscript received 18 May 1982
The thermal conductivities of some silicate glasses have been investigated by a steady-state method at 30°C. Experimental results show that the thermal conductivity progressively decreases with the increase of the formula weight of the R20 component. The effects of composition and structure on thermal conductivity are discussed.
1. Introduction M a n y of the practical applications of glasses in glass solders and glass coatings require accurate knowledge of the thermal conductivity. Heat transfer properties are also of great importance in glass melting, annealing and forming. Several authors [1-8] assumed that this property could be considered as a structure-sensitive property that responds strongly to variation in the chemical composition of the glass. A similar contribution to thermal conductivity of glass was shown by titania [5,7], lime [8] or lead oxide [9]. It was assumed that the thermal conductivity decreased when the composition of the glass was made more complex as a result of the shortening of p h o n o n mean free path because of the increased disorder of the glass structure.
2. Experimental procedure 2.1. Glass p r e p a r a t i o n
The glass compositions are reported in tables 1 and 2. The batches were prepared from Analar and reagent grade carbonates of sodium, calcium, 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
166
M.M. Ammar et al. / Thermal conductivity of silicate glasses
strontium, barium, magnesium and zinc oxide. Silica was introduced as pulverized and purified quartz of high purity. The batches were melted in platinum crucibles using an electric furnace with SiC heaters. The melting temperature was 1400°C. To ensure homogenization the melt was stirred from time to time. The melt was cast into discs which were annealed, ground and polished to a smooth flat parallel surface. The discs were between 18 m m and 5 m m in diameter and 5 m m height.
2.2. Thermal conductivity measurements Thermal conductivity measurements were carried out using an apparatus which has been described in previous articles [5,8]. In this method heat from a heat source flows through the sample under test to a heat sink which transfers this heat by thermal radiation to a surrounding evacuated chamber which has the internal surface blackened. The temperature TO of the heat source is kept constant at T t > TO during the experiment. The emissivity C of the heat sink was measured by using a standard glass sample of known thermal conductivity, and was found to be 0.94. After thermal equilibrium is reached the temperatures of the heat source, the heat sink, and the outer chamber are measured by the attached thermocouple using a Pye precision decade potentiometer to measure the thermal e.m.f, with 0.2 ~tV precision, i.e. the temperature can be measured to within 0.005°C.
2.3. Density and refractive index measurements The density of the glass samples under investigation was measured by the Archimedes method using xylene at 25°C as the immersion liquid. The refractive index was measured with an Abbe refractometer. A thin film of monobromonaphthalene liquid was used as a contact layer between the prism of the refractometer and the test glass sample.
3. Results
The data obtained are listed in tables 1 and 2. Each value of thermal conductivity is the average of at least six determinations. The uncertainty is about + 2%. From table 1 it is seen that there is a change in the thermal conductivity with a change in the content of any of the divalent metal oxides MgO, CaO, SrO, or BaO in the base soda silica glass. The glass containing MgO has the highest thermal conductivity and that containing BaO has the lowest. This result agrees with the findings of previous authors [2,3,5,8] that the oxides with the lowest molecular weight, will generally have the highest contribution to the thermal conductivity. The density and refractive index of the glass increases with the addition of BaO more than SrO and the latter more than CaO.
M.M. A m m a r e t al. / Thermal conductivity of silicate glasses
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M.M. A mmar et al. / Thermal conductivity of silicate glasses
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From table 2 it is shown that the progressive addition of ZnO to the lithium oxide glass increases the thermal conductivity, density and refractive index of the glass.
4. Discussion 4.1. Theoretical considerations
In a nonmetal heat is conducted by means of the thermal vibration of the atoms [11]. In a simple metal this mode of heat transport makes some contribution, but the observed thermal conductivity is almost entirely due to the electrons. It does not follow that the thermal conductivities of the two types of solid must be very different in magnitude, as they are in the case of electrical conductivity, but their dependence on temperature and on the imperfections in individual specimens are rather different. The process of thermal energy transfer is considered to be a random process [ 12]. The energy does not simply enter one end of the specimen and proceeds directly in a straight path to the other end, but the energy diffuses through the specimen, suffering frequent collisions. Heat is considered as being transmitted by phonons, which are the quanta of energy in each mode of vibration, and the mean free path is a measure of the rate at which energy is exchanged between different phonon modes. In terms of phonons, a flow of heat implies that the phonon distribution differs from that in thermal equilibrium, which corresponds to no net heat flow. The conductivity depends on the extent to which the distribution can depart from equilibrium for a given temperature gradient. The departure from equilibrium, which is expressed in terms of relaxation times or mean free paths, depends, in general, on the temperature and on the frequency and polarization of the phonon mode. It is the random nature of the conductivity process that brings the temperature gradient into the expression for the thermal flux. The thermal conductivity values of several noncrystalline solids are comparable and assumed to be practically identical [ 11-13]. Some authors [ 13,14] believed that there is a complete lack of sensitivity to composition in the noncrystalline solids. This would be understandable if the thermal conductivity values were as low as they could possibly be, corresponding to a diffusion of the vibration energy from one atom to the neighbouring ones. But this is not the case. The phonon mean free path is determined principally by two processes [12], geometrical scattering and scattering by other phonons. If the forces between atoms are purely harmonic, there will be no mechanism for collisions between different phonons, and the mean free path will be limited solely by collisions of a phonon with the crystal boundary and by lattice imperfections. With anharmonic lattice interactions there is a coupling between different phonons which limits the value of the mean free path.
170
M.M. A mmaret al. / Thermal conductivity of silicate glasses
The mean free path in silica glass at room temperature is 8 ,~, which is of the order of magnitude of the dimensions of a silicon dioxide tetrahedron SiO4 (7 ,~) [11-12]. The accepted concept of the nature of the glassy state [15-17] pictures silica glass as an example of a random, but continuous, network of silicon-oxygen bonds. The effective glass network size is only of the order of a single tetrahedron of the stucture. It is expected, therefore (except at very low temperatures where the phonon wavelengths are long), that the phonon mean free path will be constant, limited by the essential structural building unit size. The decline in the thermal conductivity as the temperature is lowered may thus be attributed to the decline in the heat capacity. Two models have been suggested to explain the thermal conductivity of glasses. In the first model [18], Klemens proposed that the scattering is caused by spatial fluctuations in the sound velocity and by a certain amount of long-range order and that this affects the transverse phonons more than the longitudinal ones. At low temperatures the heat is mainly carried by longitudinal phonons. As an alternative model, it is further proposed [19] that in noncrystalline solids the heat-carrying plane-wave phonons are resonantly scattered by localized phonons in a way similar to the resonant scattering found in crystals containing substantial molecules with low-frequency quasirotational degrees of freedom or in crystals containing defects causing resonant modes [20]. These two modes suggest that the scattering must have a very simple origin, quite independent of structural details or the vibrational spectrum of the solids. At high temperatures, the mean free paths for crystalline and vitreous materials approach the order of a few angstroms, i.e. the interatomic separation, as is to be expected. From the previous considerations, it seems logical to suggest that the thermal conductivity must decrease with increase of the disordering of the glass network structure and this results in a shortening of the phonon free path. The same conclusion has been arrived at by Muratov and Chernyshov [6]. To understand and interpret the effect on the thermal conductivity values of adding various monovalent or divalent oxides, the effects of such oxides on the geometrical arrangement of the building units of the glassy network are considered.
4.2. Interpretation of the results When an alkali or alkaline earth oxide reacts with silica to form a glass, the silicon-oxygen network is broken up by the alkali or alkaline earth ions as evidenced by the much lower viscosity of these glasses compared with fused silica [21,22]. So long as the number of A 2 0 or AO units, A being the metallic ion, is in less than a one-to-one ratio to the number of SiO 2 units, the silicon-oxygen tetrahedron is linked to at least three other tetrahedra, and the glass-forming tendency of the mixture is retained. As more alkali or alkaline earth oxide is added beyond the one-to-two ratio, the network becomes more and more disrupted, as more tetrahedra are bonded
M.M. A mmaret al. / Thermal conductivity of silicate glasses
171
to only two other tetrahedra. Glass formation becomes progressively more difficult because the rates of nucleation and crystallization in the glass become much more rapid as alkali or alkaline earth oxide is added. In the composition range between the disilicate (A 20, 2SiO 2 or A0.2 SiO 2) and metasilicate (A20, SiO 2 or A 0 ~. SiO2) chain-like structures should exist, although direct evidence of them is lacking. At alkali concentrations above the metasilicate isolated islands and rings; as well as chains of silicon-oxygen tetrahedra, should be the major structural features. Qualitatively it seems probable that the alkali ion not only causes a loosening of the network, but also introduces weak bonds [23]. With respect to the divalent oxides, the results can be explained by considering the type and quantity of the introduced cations on the same basis as considered for other physical properties [24,25]. These divalent cations can be present in glasses as network-modifiers within interstices or in some special oxides, as network-building tetrahedra groups. Various investigators reported that the zinc cation could have an oxygen ligancy of four or six, or both in glasses [26-30]. The substitution of ZnO for Li20 on a mole basis keeps the number of silicons and oxygens constant but reduces the number of cations per anion, allowing greater freedom for the transverse oxygen vibration [30]. Our data suggest that the zinc cations in such glass compositions show marked preference for the tetrahedral site. This is in agreement with previous assumptions [30].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16 [17] [18] [19]
G,W. Morey, The Properties of Glass (Reinhold, New York, 1954) pp. 217-20. E.H. Ratcliffe, Glass Technol. 4 (1963) 113. P.F. Van Velden, Glass Technol. 6 (1965) 166. D.G. Holloway, The Physical Properties of Glasses (Wykeham, London, 1973) pp. 41-48. M.M. Ammar, Kh. El Badry, M.R. Moussa, M.M. Halawa and S. Gharib, Cent. Glass and Ceram. Res. Inst. Bull. 22 (1975) 10. Muratov and A.V. Chernyshov, Sov. J. Glass Phys. Chem. 5 (Engl. Transl., Consultants Bureau, New York, 1979) p. 100. N.A. Ghoneim, Sprechsaal, 113 (1980) 610. N.A. Ghoneim, A.F. Abbas, M.M. Ammar and M.M. Halawa, Sprechsaal 114 (1981) 293. H.A. E1 Batal et al. Cent. Glass and Ceram. Res. Inst. Bull. (in press). G. Haacke and D.P. Spitzer, J. Sci. Instrum. 42 (1965) 702. R. Berman, Thermal Conduction In Solids (Clarendon Press, Oxford, 1976). C. Kittel, Introduction to Solid State Physics, 2nd Ed. (Wiley, London, 1962) Chap. 6, p. 138. R. Berman, Phys. Rev. 76 (1949) 315. R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. W.H. Zachariasen, J. Amer. Chem. Soc. 54 (1932) 3841. J. Biscoe and B.E. Warren, J. Amer. Ceram. Soc. 21 (1938) 287. R.H. Doremus, Glass Science (Wiley, New York, 1973) pp. 23-43. P.G. Klemens, Proc. R. Soc. A208 (1951) 108. P.G. Klemens, in: Non-crystalline Solids, Ed. J.A. Prins (North-Holland, Amsterdam, 1965). p. 162.
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M.M. A mmar et al. / Thermal conductivity of silicate glasses
[20] R.O. Pohl, in: Localized Excitations in Solids, Ed. R.F. Wallis (Plenum Press, New York, 1968) p. 324. [21] R.H. Doremus, Ann. Rev. Mat. Sci. 2 (1972) 93. [22] R.H. Doremus, Glass Science (Wiley, New York, 1973) pp. 23-31. [23] J.E. Stanworth, Physical Properties of Glass (Oxford University Press, Oxford, 1950). [24] H.A. El Batal et al. Ceram. Bull. Amer. Ceram. Soc. (1982) (in press). [25] Kh. E1 Badry et al. Sprechsaal (1982) (in press). [26] E.M. Levin and S. Block, J. Amer. Ceram. Soc. 40 (1957) 95. [27] A.G.F. Dingwall and H. Moore, J. Soc. Glass Technol. 37 (1953) 316. [28] C. Hirayama, J. Amer. Ceram. Soc. 44 (1961) 602. [29] A.R. Khan and H.E. Simpen, Glass Ind. 31 (1950) 407, 408, 428. [30] J.C. Hurt and C.J. Philips, J. Amer. Ceram. Soc. 53 (1970) 269.