THERMAL E X P A N S I O N OF SILICA AT LOW TEMPERATURES G. K. WHITE
Commonwealth Scientific and Industrial Research Organization, Sydney, Australia Received 22 October 1963
A G R E A T deal of experimental research has been done on the thermal expansion of various glasses at normal and elevated temperatures (see, for example, review by Stevelsl), and has shown that the expansion is quite sensitive to composition and to the thermal history of the glass. Behaviour at low temperatures is less well known, although it has been well established that vitreous silica has a negative coefficient of thermal expansion below 200 ° K. 2 Gibbons 3 has confirmed that this remains negative down to below 10° K. For commercial glasses such as borosilicates or soda-lime, few measurements have been made below 80 ° K; only those of Head and Laquer 4 on a Pyrex glass appear to have shown the existence of a negative expansion below 50 ° K. We now know that the 'anomaly' of negative thermal expansion .is not confined to glasses but is observed, in all the diamond-structure solids which have been investigated, including germanium and siliconP In glasses, at least in vitreous silica, there are other anomalies, such as positive temperature coefficients for the elastic stiffness moduli, negative pressure coefficients of these moduli, a large ultrasonic absorption, and a specific heat at low temperatures which is far in excess of that expected from the elastic constants. The interdependence of these anomalies has been discussed in some detail by Anderson and Dienes. 6 Considered in terms of the network hypothesis, they lead to interesting speculation concerning the particular vibrational modes which may be chiefly responsible for negative expansion, excess heat capacity, etc. In at least one crystalline form of silica (hexagonal u-quartz), these other anomalies do not exist despite the fact that the crystal is probably built up of rather similar tetrahedrally-bonded SiO4 units. The thermal expansion behaviour of crystalline forms of silica, however, has not previously been studied below liquid air temperatures. With the aid of expansion measuring equipment of,very high sensitivity,7 the author has extended earlier expansion data on pure vitreous silica, some commercial glass, and a crystalline silica (~-quartz) down to about 2 ° K. The data provide interesting 2
comparison both among themselves and with our knowledge of diamond-structure solids. A brief report of the results on vitreous silica and borosilicate was made to the Eighth International Conference on Low Temperature Physics. s Experimental method Each specimen was about 5 cm long and 2 cm diameter, and its end-faces were ground and lapped to a flatness and parallelism of 10-s cm. A conducting silver film was evaporated on to the ends and sides, and it was mounted in a copper expansion cell where one end=face formed part of a threeterminal capacitance with a parallel copper plate about 0.02 cm away (see White 7 for details). The capacitance so formed was generally in the range 5-10 pF and could be measured to 10-6 pF, so that changes in the length of sample relative to copper of ,,~ 10-9 cm could be detected. The copper cell had been calibrated with samples of A1203, MgO, germanium, silicon, and copper, each of which had also been measured in another (absolute) cell. Above 11 ° K, temperatures were measured by a Meyerstype platinum thermometer, and below 11 ° K by a helium gas thermometer and/or a germanium thermometer kindly donated by the Minneapolis-Honeywell Regulator Company, Minnesota, through Dr. Olin S. Lutes, and by their Sydney associates, Liddle and Epstein Pty Ltd. The accuracy of the measurements was such that random errors in the length change were generally less than 1 per cent or less than 0.5 A, whichever is the greater. This implies that linear expansion coefficients larger than about 10-~ deg.K -x could normally be determined with a random error of not more than 1 per cent. Values of ~ above 6 or 7 ° K were calculated from the length change produced by a temperature change of about 1° K and should not be in error by more than 1 or 2 per cent. Below 6 ° K, values were generally obtained by graphical differentiation of the length versus temperature curve. Specimens were as follows. CRYOGENICS.
(1) Spectrosil 'B' kindly given by the Thermal Syndicate Ltd (Northumberland, England), through Dr. G. E. Stephenson. This is a synthetic silica made from silicon compounds by hydrolysis or oxidation and usually contains about 0.1 per cent (wt) hydroxyl but less than 1 p.p.m, of other impurities. 9 It may be expected to have a 'fictive' temperature of about 1,050 ° C (Dr. G. Hetherington, private communication) and linear expansion coefficient ~ at room temperature of about 0.5 × 10-6 deg.K -1. (2) Vycor from Corning Glass Works, New York (rolled sample No. 7913). Vycor glass is normally made by leaching a borosilicate glass and later heating to about 1,200 ° C. The final product contains approximately 96 per cent SiO2, 3 per cent B2Oa, and a total of less than 1 per cent A1203, Na20, etc. 1° At room temperature e is about 0-75 x 10-6 deg.K -1. (3) Borosilicate. This was a heat-resistant borosilicate glass from Crown Crystal Glass, Sydney, Australia. The overall contraction in cooling from 293 ° K to 90 ° K approximated to that observed by Head and Laquer for a Corning Pyrex. Such glass usually contains about 80 per cent SIO2, 12-15 per cent B203, 4 per cent Na20, and 2 per cent A1203, and 0c at room temperature is 3 x 10-6 deg.K-1. (4) Soda-lime No. 0081 from Corning Glass Works, N.Y. This was annealed by us at 520 ° C for some hours. The Corning Company lists similar glasses as containing 72 per cent SIO2, 15 per cent Na20, 9 per cent CaO, 3 per cent MgO, and 1 per cent AI2Os, and quote ~ at room temperature as 9 x 10-6 deg.K-1. (5) and (6)Quartz, parallel and normal to the optic axis. These samples were cut from a large natural crystal of optical quality. The optic or hexad axis of the crystal was kindly determined for us by Messrs. B. Dalley and J. Kirby, of the Crystal Section of Amalgamated Wireless (Australasia) Ltd. Results
co II/I ' t I ,,"1
Quartz(i)7/,,-~'" Soda-lim..~>.~"" ~ ' q ~-~-'~'~ ~ "
. . . . . •x,,-.~ ov
F E B R U A R Y 1964
Q:er tz qi)
Z × 0
E - 200 +4
-4 10 I
. x I
Temperature(°K) • Spectrosil
Figure 1 shows firstly that the thermal expansivity (expansion per unit length) of such glasses as pure silica, Vycor, or borosilicate is both negative and large in magnitude compared with the expansivity of common crystalline solids of similar characteristic temperature 0, e.g. quartz or copper. Secondly, there is close agreement between pure vitreous silica and Vycor but the addition of soda and/or lime alters the behaviour considerably. The fact that 3 per cent B2Oa (in Vycor) has so little effect on the expansion suggests that the change observed in borosilicate is due primarily to the networkfilling agents, namely Na20, which is present probably to the extent of about 4 per cent. The soda-lime glass contains about 15 per cent Na20 and 9 per cent CaO, and its expansion differs from the pure silica by much more than does the borosilicate. CRYOGENICS.
This soda-lime glass has a small negative expansibn coefficient below 14° K, as shown in the inset of Figure 1. Quartz in the direction parallel to the optic or hexad axis also has a small negative coefficient at these temperatures but its volume coefficient remains .positive.
Figure 1. Thermal expansivity of glasses and quartz; carve for copper 7.s is included for comparison (Generally experimental points are omitted for the sake of clarity)
Figure 2 contains the linear expansion coefficients for the Spectrosil, Vycor, and borosilicate, together with values of Gibbons a for vitreous silica. It is uncertain whether the difference of about 10 per cent between Gibbons' results and the present values at 20 and 30 ° K is due to experimental error or to a difference in the equilibrium (or 'fictive') temperature of glasses. In view of the agreement between Spectrosil and Vycor it is unlikely that chemical differences are responsible, but physical differences arising from different thermal treatment cannot be excluded. 3
It is well established that the expansion characteristics above room temperature depend strongly on thermal history. TM F o r example, a glass which has been stabilized at 1,000 ° C behaves very differently from one stabilized at 1,300 ° C, but whether this persists at low temperatures is not known. Westrum t2 (see also reference 13) has determined the heat capacities at low temperatures o f two silica samples, annealed respectively at 1,070 and 1,300 ° C, and found a slight but significant difference o f a few per cent.
modes as the 'excess' heat capacity, and is it a necessary corollary to the glassy state? (2) W h y is the expansion o f Vycor the same as pure silica while those o f borosilicate and a soda-lime glass, particularly the latter, are substantially different? (3) Is the behaviour o f quartz unusual or does it conform to that of other hexagonal materials ? Before discussing these questions, consider briefly the thermodynamic relationship o f thermal expansion to other physical parameters and the behaviour patterns which have so far been observed in crystalline solids. If fl and XT are respectively the volume expansion coefficient and isothermal compressibility, then it follows that # / z ~ = (~Sl~
. . . (1)
where S is entropy, V is volume, and Cv is the specific heat at constant volume; 7 is the Grueneisen or anharmonicity parameter, which is a weighted mean of the individual 7t which express the volume dtpendence o f the frequencies vL o f the lattice modes
7i = - d i n
. . . (2)
and 3N 25
I , ,~,1,,,,1~.1 I 10 15 20 25 30
Temperoture(°K) Figure 2. Linear expansion coefficients for Spectrosil ( • ) , Vycor ( x ), and borosilicate ( ~ ). Values marked (3 are from Gibbons a
F o r T < 8 ° K the linear coefficients of three of the samples may be represented as follows = - 3 0 x 1 0 - n T 3 deg.K -~ (Spectrosil, Vycor) c~ = - 2 0 x 1 0 - n T a deg.K -~ (borosilicate) For soda glass and quartz no such simple expression is adequate. The data obtained for quartz crystals at 283, 85, 75, and 65 ° K (see Table 1) agree quite well with those listed in the N.B.S. tables; 2 for example, at 283 ° K ~01 was observed to be 7-50 x 10 -6 (cf. 7.2 x 10 -6 at 280 °, 7.5 x 10 -6 at 290 ° K), and 0~. to be 13-7 x 10 -6 (cf. 13 x 10 -6 at 250 °, 14 x 10 -6 at 300 ° K). Discussion These results raise several questions. (1) The reason Q for the very large negative expansion coefficient in pure vitreous silica. Does this arise in the same way as the negative volume expansions o f the diamondstructure solids, or from the same group o f lattice '4
the weighting factors C~ are the contributions o f each mode to the heat capacity. F o r an ideal Debye solid, the frequency spectrum is a function f(v) = av z, cut off at a m a x i m u m Vm = kO/h. Then 7, = 7 = - d i n vm/dln V is a constant. In a real solid we expect 7 to be sensibly constant at high temperatures (T >/0 where 7 = 700) and at very low temperatures ( T ~ O where 7 = 7o = - d l n 0 o / d In V) (see, for example, Barron14). The experimental curve for copper in Figure 3 illustrates this typical behaviour, and also shows that 7 ° 0 - 7o < 0 . 3 , as Table 11Linear Thermal Expansion Coefficients (× 10s deg.K-X). Value marked * is extrapolated. Those in brackets have been quoted in commercial literature or tables for glasses of similar composition T=K 283 85 75 30 25 20 15 12 10 8
Spectro- V),cor Borosil silicate (50)
- 72* --66 --59 --45 --33 --23.5 --14.3 --6"8 --2-0
--68 --64 --58
--45 --33 - 24 -14-5 -6"4 --1"8
--20 --27 -24 -19 -15 -9.5 -4.5 -I-3
Soda (90O) --+ 62 +37 +17 +4"0 +0.5 --0-5 --0-7 --0"2
Quartz Quartz Quartz (1) (H) Average 1370 640 570
750 260 225
160 103 53 17"5 6"9 2.7 1-0 0-4
, I i !
49 30-5 13'3 3"0 0-0 --0-6 --0'5 --0"25
:1160 510 455 123 79 40 13 4'6 1.6 0"5 0-2
Barron suggested from a simple model of a facecentred cubic lattice. For isotropic crystals individual )'i may be calculated from observed pressure derivatives of the elastic constants, c~
y~ = (b In cd~P):r/2zT" - 1/6 Potassium chloride is an interesting case as dc44/dp is negative, and Sheard ~s (see also Collins ~6) has shown that 7i for shear waves travelling in the (lO0) or (llO) directions is - 0 . 8 . In addition, the shear stiffness c44 is so small that these negative ~,~.values
Q; E 0r~
Cu ~ , , , , ~
~ u o r t . z
~- """ °/
Q~ - 4 L..
r/o Figure 3. Variation o f y = flV/Cx with reduced temperature 7"/0. Values o f 0 are 340 ° K (copper), 600 ° K (quartz), 380 ° K (germanium), 310 ° K (borosilicate), and 495 ° K (Spectrosil). Other data are f r o m present work and the following sources. Copper: yT.a. Germanium: y22; yo f r o m elastic constants le,~. Quartz: Cl~-; x la. Borosilicate: Cl°; X"z°. Spectrosil: Cl~'.la; X~'7
are heavily weighted at low temperatures, i.e. at the low end of the frequency spectrum. This is not quite sufficient to make the mean y negative but it does reduce it to a value of y0 ~ 0.3 compared with yoo-~ 1-5. Thermal expansion observations have confirmed this. 21 Barron TM and Blackman ~7 have both shown theoretically that the mean y might be negative under some CRYOGENICS-
circumstances, if some yi for shear modes are negative and if they are weighted sufficiently in the average. Blackman concluded that ' . . . the calculations suggest that negative volume expansion coefficients are to be expected from open structures rather than closepacked "ones and that those with relatively low shear moduli would be favoured . . . . ' Barron has given a simple mechanical illustration of why excitations of transverse or shear modes tend to contract the lattice rather than to expand it. Diamond-structure solids might be expected to display such negative expansion. They are of open structure, with each atom having only four near neighbours arranged tetrahedrally. Indeed, thermal expansion observations have shown that in all cases examined (silicon, germanium, lnSb, c~-tin, CdTe, GaAs, ZnSe, diamond) ~ does become negative below 50-100 ° K. It has also been observed more recently that for germanium and silicon at the lowest temperatures c~ becomes positive again and approaches the limiting value calculated from elastic data. 22,23 (See Figure 3). The negative expansion has not yet been explained satisfactorily but it is almost certainly connected with the existence in these solids of a .highly dispersive, transverse acoustic branch of the frequency spectrum which is fully excited below 100~ K 24 and for which the higher frequency )'i values may well be negative. Turning to silica glass, consider the random network model of Zachariasen and Warren (Figure 4). X-ray and neutron scattering indicate short range order wherein each silicon atom is bonded to four oxygen atoms; the average Si--O distance is about 1.6 A and Si--Si distance is 3-0 A, the Si--O--Si bond angle being about 150L The ring of atoms making up the network generally contains six silicon atoms although probably a significant proportion has five or seven. The open nature of the structure may be judged from the density which is about 2.20 g/cm 3, considerably less than that of c~-quartz, for which p = 2.65 g/cm 3. ~-quartz has a hexagonal structure formed also of SiO4 tetrahedra linked through common oxygen atoms. The addition of borate or alumina to silica results in some of the silicon atoms in the network being replaced by boron atoms or aluminium atoms. However, monovalent or divalent additives such as sodium, potassium, or calcium go into the interspaces in the network t (Figure 4). The negative thermal expansion in vitreous silica is much larger in magnitude and persists to a much lower temperature (T-~ 0/200) than in germanium or silicon. Figure 3 shows that ), reaches a value of about - 8 near 3 ° K, and the limiting value at low temperatures calculated (by Dr. J. G. Collins using his previous averaging procedure ~6) from the elastic data of Anderson 27 is ~'0 = -2"36. It seems improbable that vibrational modes of the silicon atoms 5
(within the SiO4 group) are chiefly responsible for this behaviour of glass. More likely is the occurrence of low frequency transverse modes of the oxygen atoms (Smyth, 2s Anderson and Dienes6). In the open structure of vitreous silica the oxygen atoms appear to be relatively free to vibrate in directions normal to that of the Si--O--Si linkage, with no near neighbours to exert a strong restoring force. This type of low frequency transverse mode would be most easily excited at low temperatures and could lead to a contraction of the lattice. Supporting evidence comes from Anderson and B6mmel, 26 who observed a broad absorption peak for ultrasonic waves centred at about 40-50 ° K, and from heat capacity observations. 12:a Even at temperatures as low as 3 ° K (T < 0/100) the measured heat capacity exceeds that calculated from elastic constants by more than 100 per cent. Thus 0o -~ 395 ° whereas 0elasUe '~ 495 ° K. Flubacher et al) 3 have shown (from Brillouin spectra) that this cannot be due to dispersion but must arise from peaks in the frequency spectrum consisting of '... optical modes of unusually low frequency. The presence of such modes is clearly indicated by the observed Raman spectrum . . . . . ' The excess
Silicon Boron,Aluminium Sodium.Calcium
Figure 4. Schematic two-dimensional representation o f a glass network t
heat capacity can be fitted assuming Einstein terms near 10 and 50 ° K and containing a total of not more than about 1 per cent of the total number of modes. 2s By contrast, hexagonal quartz has a normal heat capacity (0o ~ 0elastic ~ 600 ° K 6:2) and its Raman spectrum shows no bands suggestive of optical medes in the low temperature region. The similar behaviour of Vycor indicates that substitution of a small percentage of boron atoms at the silicon sites has little effect on the modes responsible for negative expansion, as might be expected 6
if they are transverse modes of the oxygen atoms. However, the addition of network-filling atoms such as calcium and sodium does have considerable effect; in the soda-lime glass about 25 per cent of (Na20 + CaO + MgO) is seen to be sufficient to almost completely suppress the negative expansion, presumably by impeding the transverse vibrations of the oxygen atoms. Crystalline quartz shows considerable anisotropy including negative values for % below 12 ° K, but the volume coefficient fl = at, + 2ctI remains positive. This behaviour resembles that of the hexagonal metals magnesium, beryllium, cadmium, and zinc (see reviewS), for which the anisotropy of expansion increases with the anisotropy of atomic packing and of elastic moduli. Magnesium is nearly isotropic in its properties, having an axial ratio c/a = 1"623 (cf. ideal packing, c/a = 1.633). Zinc has an axial ratio c/a = 1-86, is much stiffer in the basal plane (a direction) than in the c direction, and has a much smaller expansion in this basal plane. The thermal vibrations in the softer c direction of zinc are excited preferentially at low temperatures and cause ~, to increase very rapidly while making ~± become and remain negative up to about 70 ° K. Quartz is more akin to beryllium (for which c/a = 1-58) as it is softer in the basal plane than in the c direction and has a correspondingly larger expansion coefficient in this softer direction. The variation of 7 with temperature in quartz (Figure 3) is a little puzzling. Experimental data on highly anisotropic crystals in different orientations and extending down to low temperatures are very meagre but there is as yet no evidence that they behave very differently from cubic materials, for which the major changes in 7 occur between 0/5 and 0/10. For quartz the apparent decrease in y below 0/30 seems to lie outside the experimental error. This form of y(T) may arise from a complicated frequency spectrum for different branches of which y vanes greatly. Before a complete picture is obtained of thermal expansion in the various forms of silica, it may be useful to examine other samples such as (1) pure vitreous silica stabilized at different 'fictive' temperatures, (2) a pure boroxide glass, (3) perhaps a nonoxide glass such as selenium, (4) the cubic form of SiO2, cristobalite, which also consists of linked SiO4 units but has a density Co = 2.30) much closer to glassy silica than does a-quartz, and (5) neutronirradiated samples of quartz. Finally, it would be interesting to know how the heat capacity and the thermal expansion of vitreous silica vary with temperature below 2 or 3 ° K as the low frequency optical modes are rapidly becoming 'frozen' (?), although it presents a considerable problem to determine the very small length changes likely to occur below 2 ° K. CRYOGENICS.
F E B R U A R Y 1964
This research was supported by the United States Air Force Office of Scientific Research. The author is also grateful to his colleagues, Dr. J. G. Collins, Dr. R. D. McCammon, and Mr. A. F. A. Harper, for their assistance and numerous discussions, to Mr. R. Abeli of the optical workshop for preparing specimens, to Mr. J. W. W. Smyth for copious quantities of liquid helium, to Mrs. R. Munsie for help with calculating results, and to Dr. E. F. Westrum, Jr. for sending his unpublished data on the specific heat of various forms of silica.
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9. HETHERINGTON,G., and JACK, K. H. Phys. Chem. Glasses 3, 129 (1962) 10. NORDBERG,M. E. J. Amer. ceram. Soc. 27, 299 (1944) 11. BROCKNER, R. Naturwiss. 7, 150 (1962) 12. WESTRUM, E. F. University of Michigan, unpublished data~ 13. FLUBA'CHER, P., LEADBETTER, A. J., MORRISON, J. A., and STOICH~F, B. P. J. Phys. Chem. Solids 12, 53 (1960) 14. BARRON, T. H. K. Ann. Phys., N.Y. 1, 77 (1957); Phil. Mag. 46, 720 (1955) 15. SHEARD,F. W. Phil. Mag. 3, 1381 (1958) 16. COLLINS,J. G. Phil. Mag. 8, 323 (1963) 17. BLACKMAN,M. Phil. Mag. 3, 831 (1958) 18. HUNTINGDON,H. B. SolM State Phys. 7, 213 (1958) 19. SMrrH, P. L., and WOLCOTT;N. M. Phil. Mag. 1, 854 (1956) 20. MOREY, G. W. Properties of Glass (2nd Ed.) (Reinhold, New York, 1954) 21. WHITE, G. K. Phil. Mag. 6, 1425 (1961) 22. MCCAMMON, R. D., and WroTE, G. K. Phys. Rev. Lett. I0, 234 (1963); CARR, R. H., MCCAMMON, R. D., and WHITE, G. K. Unpublished 23. DANIELS,W. B. Phys. Rev. Lett. 8, 3 (1962); SCHUELE,D., and SMITH, C. S. Case Institute, Cleveland, unpublished (1962) 24. BROCKnOUSE, B. N., and IYENGAR, P. K. Phys. Rev. 111, 747 (1958) 25. SMYTH, H. T. J. Amer. ceram. Soc. 38, 140 (1955); 36, 327 (1953.) 26. ANDERSON,O. L., and B6MMEL, H. E. J. Amer. ceram. Sac. 38, 125 (1955) 27. ANDERSON,O. L. Progress in Very High Pressure Research, p. 225 (Ed. Bundy et al.) (Wiley, New York, 1961) 28. See also CLARK, A. E., and STRAKNA, R. E. Phys. Chem. Glasses 3, 121 (1962); LEADBETTER,A. J., and Mol~a~mON, J. A. Phys. Chem. Glasses 4, 188 (1963)