The linear thermal expansion of Zerodur has been measured from 20 to 300 K and the thermal conductivity from 2 to 100 K. For each property the temperature dependence appears to reflect the composite nature of the ceramic-glass.
Thermal properties of Zerodur at low temperatures R.B. Roberts, R.J. Tainsh and G . K . W h i t e Key words: cryogenic, Zerodur, thermal properties
Zerodur is a glass-ceramic made by Schott I and designed to have a near-zero expansion coefficient at room temperature which is important in constructing large optical components. This expansion is achieved by careful adjustment of the constituents and suitable heat treatment which partially devitrifies the glass and produces a crystalline phase containing some components of negative expansion coefficient, for example/~-eucryptite (Li20. A12O3. SIO2).2 Some of these crystals can also be considered as a substituted form of t3-quartz which itself has a negative coefficient. The glassy matrix may constitute 20 or 30% by volume and presumably has a small positive coefficient, although glasses do exist which have a negative coefficient at room temperature, eg a titanium silicate such as Corning ULE. During 1972-743 we measured the linear expansion coefficient a of samples of Zerodur, Cer-Vit (Owens Illinois Inc.), ULE, and found the behaviour of each was qualitatively similar to vitreous silica below 100 K, iea was negative, varied roughly as T 3 below 10 K, and reached a minimum between 30 and 50 K. However, whereas a (T) for vitreous silica and ULE increased monotonically from about 60 K to room temperature, there was evidence that a(T) for Cer-Vit and Zerodur reached a maximum (and perhaps became positive) between 100 and 300 K. Here we report measurements on Zerodur of a from 2 to 300 K and of the thermal conductivity ~ from 2 to 100 K. These were stimulated by inquiries from O. Lindig of Schott 1 and Dr R. Schlegelmilch of Carl Zeiss (West Germany) concerning thermal property data needed in the design of space telescopes. E x p e r i m e n t a l details
platinum thermometer). Values of ~1/l at intervals of 2.5 K were fitted to a fifth-order polynomial with arms deviation of < 10 -7 and a values obtained by differentiation. Intercomparison with silicon 6 indicates that errors in a should not exceed 10-8 K-1 .
Thermal conductivity from 2 to 100 K. This was measured on a cylinder 60 mm long and 10 mm in diameter in a longitudinal heat-flow steady-state cryostat similar to that described by White. T Temperature gradients were measured down a 10 mm section using a pair of germanium thermmeters from 2 to 30 K and platinum thermometers from 30 to 100 K. They were calibrated on IPTS-68 Temperature Scale by the CSIRO Division of Applied Physics Thermomerry Group. Judging from reproducibility, random errors in X values are less than 2% but systematic errors may reach 5%. The latter arise from heat leakage along leads which is significant in materials of low heat conductivity. Results
Thermal Expansion. In Fig. 1 are shown smoothed values of a from present measurements together with values obtained by Lindig 8 above 100 K on a sample from the same melt. For comparison we include values from a different sample (first measured here in 1972) below 130 K; they agree within experimental error below 30 K but are significantly different from 50 to 120 K. At liquid helium temperatures data on both samples are represented within 10% by a T 3 relationship: For present Zerodur, a = - (1.45 -+ 0.10) × 10 -1° T 3 K -1 (4 < T < 10 K) I00 /
Measurements were all made on samples taken from melt Number 890162 - K28975 supplied by O. Lindig.
Thermal Expansion from 2 to 125 K. This was measured in a three-terminal capacitance dilatometer 4 on a cylinder 50.8 mm long and 20 mm in diameter. The end-faces of the cylinder were lapped flat and parallel and then silvercoated by evaporation. Random and systematic errors in the values of a should not exceed 3 x 10-1° K -1 at liquid helium temperatures, 10 . 9 K -1 at 10 K, 10 -8 K -I at 35 K and 2 x 10 -8 K-1 at 100 K.
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The authors are at CSlRO Division of Applied Physics, Sydney, Australia, 2070. Paper received 1 June 1982. 0011-2275/82/011566-03 566
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Thermal expansion from 80 to 320 K. Measurements were made on a 50 mm long polished block using a polarisation interferometer s with length resolution of 0.3 nm and temperature resolution of 1 mK (using a small Rosemount
I00 150 Temperature, K
Fig. 1 Linear Coefficient of Thermal Expansion c~(T) for present Zerodur (O -- capacitance dilatometer, X -- interferometer, and
• -- Lindig7) compared with earlier Zerodur sample ( - - ) , silicon crystal, 6 vitreoussilica (aged 1400°C), 3 ULE 7971 glass3,5
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compared with White, s a = - (1.49 +- 0.06) x 10 -1°
We analysed the a-values from 2 to 6 K carefully to see whether there was a departure from T s indicative of a T-term which has been observed in the heat capacity and expansion coefficient of some glasses and attributed to tunnelling. Leadbetter et al9 have observed a marked departure of Cp from T s dependence in Cer-Vit below 4 K which if it were similarly present in Zerodur might give rise to an observable T-term in a. Our data indicate that any T-term is less than 10-10 T in this Zerodur compared with ca - 3 x 10 -1° T in vitreous silica and ULE and - 3 x 10 .9 T in PMMA. 10 Also shown in Fig. 1 are curves of a(T) for a diamondstructure crystalline solid, silicon, 6 and two glasses with negative coefficients at low temperatures, namely pure vitreous silica s and ULE 7971.s's Smoothed values are given in Table 1.
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Thermal conductivity. Smoothed values of X are given in Table 1 and raw data are plotted in Fig. 2. They seem consistent with the room temperature values obtained at the Physikalische Technische Bundesanstalt (Braunschweig, FRG), 11 but are significantly larger than the low-temperature values from the Linde Company (IYdllriegelskreuth, FRG). it
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Temperature, K Table
values for Zerodur
1 0 - 8 K -1
W m -1 K -1
Fig. 2 Thermal Conductivity X(T) for Zerodur (e -- present, o - Linde, • -- PTB) compared with vitreous silica, 12 polvcrvstalline alumina 13 (grains 5 - 30 #m) and "estimated" values for very fine grained quartz (50 nm). Note that the vertical scale for alumina is changed by factor of ten
Also included for comparison are representative curves for a glass (vitreous silica 12) and fine-grained polycrystals of alumina 13 and quartz. Discussion
Thermal expansion. Zerodur is a composite material of crystalline phases set in a glassy matrix. According to the manufacturer, crystals average 500 A (50 nm) in size and occupy about 70% of the material. The crystalline phase consists of a solid solution of 13-eucryptite with main constituents Li20, A1203, P2Os and SiO2 which is known to have a negative coefficient of expansion at room temperat u r e ) Unfortunately little is known of the behaviour of a(T) of individual silicates at low temperature. We do know the following three points: One, many crystals of low coordination number such as sphalerite, wurtzite, chalcopyrite, cuprite, have negative values of a below 50 or 100 K (see silicon in Fig. 1) which arises from the strong influence of low-frequency transverse acoustic (TA) modes of vibration 1° two, both a-quartz and cristobalite forms of SiO2 have average a-values which are positive at all temperatures; and three tetrahedrally bonded glasses of SiO2, GeO2, BeF2 and also Zn(PO3)2 have negative a(T) at low temperatures. 14 Fig. 1 shows a(T) for pure vitreous SiO2 and Coming ULE (SiO2 + 7.5% TiO2) to fall much more sharply below 20 K than for silicon and then rise steadily above 60 K. Small amounts of network modifiers such as B2 03, TiO2 do not change a (T) dramatically but fillers such as Na20, K20 etc, increase a, ie make it more positive. 1° Turning to Zerodur, we suggest that the rapid decrease in a from 2 to 30 K is chiefly due to the glassy matrix. Above 30 K, a rises with a plateau or knee between 100 and 200 K. This knee may be due to an increasing negative contribution from some silicate crystals in competition with a slow rise
in ot of the glassy phase. The Cer-Vit which we examined earlier 3 showed similar behaviour to Zerodur below 100 K. Many Cer-Vit samples were measured by Berthold and Jacobs is from 150 K upwards and most showed positive ot values between 150 and 250 K indicating a much more distinct maximum than in the present Zerodur. The distinct differences in this temperature region from specimen to specimen must reflect small differences in composition and the sensitive balance between positive and negative components. Thermal conductivity. The conductivity at low temperatures behaves qualitatively like a composite of crystallite and glass. X(T) shows an inflection rather than the plateau (cf vitreous silica in Fig. 2) exhibited by most glasses but varies with temperature less rapidly than crystals (cf finegrained alumina in Fig. 2) when grain boundary scattering limits the paths of the phonon heat carriers. Measurements of X(T) for epoxy matrices containing micron-sized particles of alumina or diamond show ?,(7") dependences similar to these. 16 We do not have conductivity data at low temperatures on very finegrained (500 A or 50 nm crystallites) silicates, but we can estimate the behaviour from ~(T) for quartz crystals of 5 mm diameter by scaling ~ down by a size factor 5 mm/50 nm = 10 s as shown in Fig. 2. We can also calculate approximate values for the mean free path l of the phonons in Zerodur assuming ~ = 1/3 Cvl where approximate heat capacity C per unit volume is taken from data on Cer-Vit 9 and average velocity v for sound waves is assumed to be 4 × 10 s cm s-1 . This leads to l = 5 to 10 A (0.5 - 1 nm) above 100 K, 20 A at 25 K, 40 A at 15 K, 100 A at 10 K and rising rapidly to ca 1000 A at 4 K and 2500 A at 2 K. Thus at T~> 20 K, behaviour is like a glass; as long as the wavelength of the important phonon modes is ~< 10 A, the mean free path is limited by the scale of the disorder in the glass which is ~ 10 h. However, as the wavelength increases on cooling below 20 K, the scattering decreases and l increases but less rapidly in Zerodur than in vitreous material. This difference may be due to the grain size ( ~ 500 A) in the crystal phase rather than boundary resistance (acoustic mismatch) at junctions between grains and glass, is'z7 We need heat capacity and ultrasonic data on the present Zerodur before we can be confident of these mean free paths. Lawless zs has also measured X(T) below 25 K on a machineable-glass-ceramic (Corning MGC) and found changed much more rapidly with T than in the present case. From 2 to 4 K he found 2~= T. 2,3 This may reflect the fact that MGC has much less glass phase, consisting
largely of interlocking crystallites (5-10/am size) of a fluorophiogopite mica. Conclusion
Measurements of thermal expansion of Zerodur have been made from 2 to 300 K by different techniques which agree within respective limits of error in the ranges of overlap. The behaviour of a(T) is complex due to the composite nature of the glass-ceramic in which some components have negative and some have positive coefficients of expansion at different temperatures. Below about 10 K, ~ "~ - 1.4 x 10 -1° T 3 with no evidence of a significant T-term between 2 and 4 K. The heat conductivity from 20 to 100 K is similar to that of vitreous silica, but from 20 down to 2 K, ?,(7") behaves in a way intermediate between glass and crystal, varying roughly as T. °'9 Below 5 K, the mean free path of the phonons appears to exceed the average size of the crystallites, but is less than in vitreous silica. We thank O. Lindig of Schott Jena for giving us samples and his unpublished data, Dr. Schlegelmilch of Carl Zeiss for his interest and Mr. C. Andrikidis and Mr. J. Seckold for preparing and mounting samples. References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Jenaer Glaswerk Schott & Gen., Mainz, West Germany McMiilan,P.W. Glass Ceramics, 2nd Ed Academic Press, London (1979) White, G;K. Cryogenics 16 (1976) 487-606 White,G.K., Collins, J.G. J Low Temp Phys 7 (1972) 43 Roberts, R.B.JPhysE 14 (1981) 1386 Lyon, K.G., Salinger, G.L., Swenson, C.A., White, G.K. J App Phys 48 (1977) 865 White,G.K. Thermal Conductivity Vol 1 (Ed. R.P. Tye) Academic Press, London (1969) 93 Lindig,O. private communication (1981)
Leadbetter, A.J., Jeapes, A.P., Waterfield, C.G., Wyeherley, K.E. Chem Phys Lett 52 (1977) 469 Barton, T.H.K., Collins, J.G., White, G.K. Adv Phys 29 (1980) 609 Sehlegelmileh,R. private communication (1981) Damon, D.H. PhysRevB8(1973) 5860 Berman,R. Proc Phvs Soc A65 (1952) 1029 Krause,J.T., Kurkjian, C.R. J A m Cer Soc 51 (1968) 226 Berthold, J.W., Jaeobs, S.F. Appl Opt 15 (1976) 2344 Garrett, K.W., Rosenberg, H.M.JPhys D 7 (1974) 1247
Berman,R. Thermal Conduction in Solids (1976) Clarendon Press, Oxford Lawless,W.N. Cryogenics 15 (1975) 273
C R Y O G E N I C S . N O V E M B E R 1982