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Thermal expansion of glass
CHAPTER 10
Thermal expansion of glass 10.1 Introduction When a glass is heated, it generally expands. If the temperature over the body of glass is equal everywhere and the body is not restrained, then there will be no development of stress in the body. If, on the other hand, there were nonuniform heating over the body, then the different layers of glass would attempt to expand differently. The enforcement of the compatibility criteria (see Appendix) results in a different set of dimensional changes, and consequently, stresses develop. The tensile stress components could lead to glass failure (which is discussed in detail in Chapter 18). The magnitude of the stresses so generated is related to thermal expansion. The expansion due to heat flow in or the contraction due to heat flow out of the body is, therefore, an important consideration in product design. Thus, architectural glass window panes need to tolerate a temperature gradient across their thickness created by the difference between room temperature and the temperature on the outside. A hot glass-baking dish should be able to withstand the “thermal shock” when set on a cold kitchen countertop. It may be realized that thermal compatibility principles apply not only to the various parts of the glass body but also to other materials which have been joined to the glass. Thus, the success of glass-to-metal seals in a variety of today’s products critically hinges on the thermal expansion of glass (and of the metal). Consideration of thermal expansion is also important in the annealing of glassware. Later in Chapter 13, we will show that annealing refers to the release of thermal stresses generated from the nonuniform cooling of glass through the glass transition range during glass forming. In this chapter, we will review the thermal expansion property of the glasses.
10.2 Definitions Over the temperature range T1 to T2, the mean or the average volume expansion coefficient βm is given by Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00010-9
© 2019 Elsevier Inc. All rights reserved.
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βm ¼
V2 V1 V1 ðT2 T1 Þ
(10.1)
where Vi is the volume at temperature Ti. The change in volume is to be measured under closed isobaric (constant pressure) conditions. The true or instantaneous volume expansion coefficient β is given by 1 ∂V β¼ : (10.2) V ∂T P The corresponding linear expansion coefficients (α) are obtained by replacing volume V by length L. Appropriate units for the coefficient of thermal expansion (CTE) are /°C in the cgs system and K1 in the SI system. Also, for small expansion coefficients, β ¼ 3α:
(10.3)
10.3 Methods of thermal expansion measurement In most practical applications linear thermal expansion is of primary interest. It is generally measured by three methods: pushrod dilatometers, interferometry, and standard glass seals. The experimental setup for a typical commercial pushrod dilatometer is shown in Fig. 10.1. The specimen, held against the flat wall of the measuring system, pushes on a horizontal spring-loaded pushrod on heating. This change relative to either the system tube itself or to a second pushrod against a standard specimen is measured using an LVDT (linear variable differential transducer). The measuring system and the pushrods are generally made out of alumina or fused silica. An equal length platinum sample is generally a good standard to use. The key to getting good data is the uniformity of temperature over the specimen and the standard; both of which should have flat, parallel ends. The differential accuracy improves if the unknown and the
Fig. 10.1 Schematics of a double pushrod differential dilatometer. (Courtesy: Theta Industries, Inc., Port Washington, New York.)
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standard are close in expansion characteristics. Some commercial differential pushrod dilatometers can be accurate to as little as 0.5% which is generally quite adequate. A typical plot from a dilatometer (after adding the expansion of the standard specimen) is shown in Fig. 10.2. Note the rapid rise in expansion which marks the onset of the glass transition range. The curved hook downwards is an artifact of the measuring technique itself. At this temperature, the expansion of the specimen is countered by an apparent contraction due to specimen softening (pushrods begin to penetrate into the specimen or the latter begins to sag because the viscosity is about 109–1010 Pa s). This temperature is called the dilatometric softening temperature Td. The measurements should be stopped soon after. (Failure to stop the furnace from heating could result in the specimen getting much too soft and possible damage to the system due to glass adherence.) The return (contraction) path is never parallel to the expansion path in the glass transition range. This is explained further in Chapter 13. For now, it suffices to suggest that, for many applications, it is the contraction curve which is important in the consideration of stress generation and not the expansion
Fig. 10.2 Effect of the cooling rate history on the expansion of a glass.
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curve. For sealing with glass, the thermal contraction data should be computed from the set point to the room temperature. The set point may be thought to be a cooling rate-dependent idealized temperature below which the glass may be considered an elastic solid. In the interferometric method, three carefully ground and polished pieces, two of a known and one of the unknown glass, separate two optically flat and parallel surfaces. As the temperature increases, the separation also increases, which can be monitored by interferometric methods [1, 2]. When a glass is to be employed for sealing purposes, a more accurate method of determining the expansion is by employing the unknown as a component in a standard seal. Convenient seal forms are the sandwich seal (described in ASTM F144-73) and the trident seal [3]. A sandwich seal is formed by fusing three plates to each other in such a manner that the outer pieces may be a known glass, and the inner piece is the unknown glass. A trident seal is a variation of the sandwich seal. Here the pieces neither have to be plates nor have to be fused to each other along the entire length. They merely have to be joined together at the ends. One then has to make a measurement of the optical retardation (generally at the center) and convert this optical retardation to a stress (see “Photoelastic Properties,” Section 19.2.6). Once the stress is known, the mismatch is calculated [4] from standard formulas. A thermal contraction coefficient may be extracted [5] from the mismatch if the temperature interval ΔT between the set point and the room temperature is known. To examine the suitability of a thick or thin glass film fired onto a ceramic, metal, or glass substrate, it is often useful to measure the deflection of composite strips or circular plates. In particular, the depression h in a circular disk of radius r, thickness ts, over which a glass coating of thickness tg has been applied can be converted to yield the differential thermal contraction coefficient Δα using the expression [6] Δα ¼
ts2 K1p h , 3ΔT tg + ts r 2
(10.4)
where K1p ¼ 4 + 6
2 3 tg tg Eg tg 1 νg 1 νs Es ts +4 + + ts ts Es 1 νg ts Eg 1 νs tg
Here, E and ν are the Young’s modulus and Poisson’s ratio; the subscripts s and g refer to the substrate and the glass coating. For glass coating thickness tg much less than the substrate thickness ts, the above formula reduces to
Thermal expansion of glass
Δα ¼
ht 2s 3r 2 ΔTt g
1 νg Es , Eg 1 νs
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(10.5)
which is also obtained [7] by extending Stoney’s formula [8] for stresses in a bimetallic disk. A semiquantitative technique used by lampworkers to determine compatibility of sealing between two glasses is the composite fiber method discussed in Exercise 8.
10.4 Thermal expansion versus composition and temperature Before studying various thermal expansion data, it should be noted that most expansion coefficients in the glass literature are quoted in terms of 10–7/°C. On the other hand, metallurgists quote the metal expansion data at 10–6/°C or ppm. (This, of course, is important during a conversation.) Yet another issue, often, is the range of temperatures over which the value of the expansion coefficient is quoted. Glass manufacturers generally quote the mean expansion coefficient over 20–300°C, which is the assumed range for most of the expansion coefficient values cited unless specified otherwise. Of all the single-component glasses, silica glass has perhaps the lowest thermal expansion coefficient. The average value between 20°C and 1000°C is roughly 5.5 10–7/°C. (Because of this rather low value of α, silica glass is preferred for products which require a high thermal shock resistance; see below.) In marked contrast to the expansion of silica glass, the mean linear expansion coefficient from 20°C for cristobalite increases from about 55 10–7/°C to about 200 10–7/°C at 200°C, and it undergoes a large expansion increase at the α ! β conversion to an effective mean expansion coefficient of 550 10–7/°C at about 270°C, subsequently decreasing to that of silica glass at temperatures above 1000°C. (The mean expansion coefficient values stated here are one-third of the measured mean volumetric expansion coefficients, see Ref. [9]). The contrast between the expansions of silica with other glass-forming oxides such as B2O3 (150 10–7/°C) and GeO2 (77 10–7/°C) is also noticeable. In the vicinity of 0 K, vitreous silica and several other tetrahedral network formers display negative expansion (Fig. 10.3). For vitreous silica, the anomaly continues [10] from 0 K to about room temperature. This anomalous behavior, similar to that for the elastic properties of silica has been explained by Vukcevich [11] in terms of changing population of SiOSi bond angle between
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Fig. 10.3 Thermal expansion curves of some single-component glasses at low temperatures. (Modified from J.T. Krause, C.R. Kurkjian, J. Am. Ceram. Soc. 51 (1968) 226.)
145° and 138° (Section 8.2.4). More on the anomalous behavior of silica at low temperatures is discussed in the following chapter on heat capacity. The addition of alkali to the fully connected silica network breaks up the oxygen bridges, resulting in a monotonically increasing thermal expansion coefficient as a function of the alkali content (Fig. 10.4A): the higher the alkali content, the larger the expansion (Fig. 10.4B). In the alkali borate glasses, however, pronounced minima in the thermal expansion coefficient are observed as a function of the alkali concentration (Fig. 10.5). This
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Fig. 10.4 (A) Thermal expansion coefficients of R2OSiO2 (R ¼ Li, Na, K) glasses as a function of composition. (B) Thermal expansion curve for xNa2O(100 x)SiO2 glasses.
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Fig. 10.5 Thermal expansion coefficients of R2O-B2O3 glasses as a function of composition.
observation, otherwise called the boron anomaly, has already been discussed in Section 5.6. B2O3 addition to SiO2 also increase the expansion coefficient monotonically (Fig. 5.18) despite the absence of nonbridging oxygens. Here, the expansion increases as a result of the freedom allowed by the triangular boron connections. The addition of TiO2 to silica lowers [12] the expansion coefficient. As seen in Fig. 10.6, a 7.5TiO292.5SiO2 (wt%) glass made by the chemical vapor deposition (CVD) process (Chapter 22) has an expansion coefficient of 0 0.3 10–7/°C over the 5–35°C range (Corning “Ultra Low Expansion” ULE glass). Such zero expansion glasses are used as honeycomb substrates for space-based lightweight mirrors. Table 10.1 lists the expansion coefficients of some commercial glasses. The soda lime silicate glasses generally have an expansion coefficient of 85–95 10–7/°C. Between the soda lime silicate glasses on one side and vitreous silica on the other side are the alkali borosilicate and the alkaline earth aluminosilicate glasses. The expansions of the alkali borosilicates range between 10 10–7/°C (Vycor) and about 55 10–7/°C (GE 706, Corning 7052), and those of the alkaline earth aluminosilicates around 45 10–7/°C (GE180).
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Fig. 10.6 Thermal expansion of TiO2-SiO2 glasses as a function of temperature. (Modified from Ref. H.A. Miska, in: Ceramics and Glasses, Engineered Materials Handbook, vol. 4, ASM Publ., 1992, pp. 1016–1020.)
Table 10.1 Thermal expansion coefficients of some commercial glasses Expansion coefficient 1027/°C Glass code Type 0–300°C Set point — 25°C
GE 001 GE 008 GE 012 Corning 1720 GE 706 GE 725 Corning 7720 Corning 7740 Corning 7913 Corning 7940
Potash soda lead Soda lime Potash soda lead Aluminosilicate Borosilicate Borosilicate Borosilicate Borosilicate 96% silica Fused silica
93.5 93.5 89.5 42 48 35.5 36 32.5 7.5 5.5
101 105 97 52 55 39 43 35 5.5 3.5
Additivity factors for the various components in the glasses have been obtained by various workers. Some of these are listed in Table 10.2. Factors provided by Lakatos et al. [13] apply closely to the commercial soda lime silicate compositions.
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Table 10.2 Additivity factor, pi, for calculating thermal expansion coefficientsa Oxide Winkelmann and Schottb English and Turnerc Gilard and Dubruld
SiO2 B2 O 3 Na2O K2O MgO CaO ZnO BaO PbO Al2O3
0.267 0.033 3.333 2.833 0.033 1.667 0.60 1.0 1.0 1.667
0.050 0.653 4.16 3.90 0.45 1.63 0.70 1.40 1.06 0.14
0.04 0.4 + 0.01w 5.1 0.0333w 4.2 0.0333w 0 0.75 + 0.035w 0.775 0.025w 0.91 + 0.014w 1.15 0.005w 0.2
P α ¼ piwi 107/°C; wi ¼ wt%. A. Winkelmann, O. Schott, Ann. Phys. (Leipzig) 51 (1984) 735. c S. English, W.E.S. Turner, J. Am. Ceram. Soc. 12 (1929) 760. d P. Gilard, L. Dubrul, Verre Silic. Ind. 5 (1934) 122. a
b
10.5 Concepts of glass expansion In ionically bonded crystalline materials, one thinks of the thermal expansion to arise from the asymmetric shape of the Condon-Morse potential well diagram. The asymmetric shape arises because of the anharmonic terms in the expression for the potential well. When the energy is raised by adding thermal energy, the ion is able to vibrate between points B and C (Fig. 10.7). The average location of the ion appears to be A0 , A00 , etc., and hence, the effective separation between the ions increases, as shown in Fig. 10.7. (The reader should note that the dissociation or sublimation energy of ions is related to the depth of the potential well, the Young’s modulus is related to the curvature of the trough, and the thermal expansion is related to the asymmetry of the potential function.) However, much like the argument presented for the elasticity in glass (Chapter 8), network structures can also expand or contract by a bondbending mechanism. Thus, in effect, the expansion of a glass may have very little resemblance to the thermal expansion of a crystal of identical composition.
10.6 Configurational versus vibrational contributions to thermal expansion The CTE of a glass-forming liquid is generally higher than that of its corresponding glass. This is because the CTE of the liquid includes contributions
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Fig. 10.7 Change in the atomic separation, r, with temperature. At 0 K, the equilibrium separation is r0. Thermal energy at T1 causes oscillations between ra and rb; the mean position re is greater than r0 because of the asymmetry of the potential function.
from both configurational and vibrational motions, whereas in the glassy state the configurational transitions are frozen so that the CTE is entirely a result of anharmonic vibrations within a single energy well. In contrast, the liquid state is highly fluid so the sample can transition between different atomic configurations at different temperatures. These different atomic configurations generally correspond to different molar volumes, and it is this change in volume that gives rise to the configurational contribution of a liquid’s thermal expansion coefficient. Hence, in a glass the CTE is equal to the vibrational contribution only, that is, αglass ¼ αvib, whereas in a liquid the CTE is the sum of vibrational and configurational contributions: αliquid ¼ αvib + αconf ¼ αglass + αconf. An example of separating the CTE of the liquid into vibrational and configurational contributions is shown in Fig. 10.8. Here the vibrational CTE (¼αglass) is obtained through standard dilatometry of a glass sample. The configurational CTE is obtained by equilibrating glass samples in a furnace at different temperatures and then rapidly quenching to room temperature to freeze in the particular configurations (i.e., to freeze in particular values of the fictive temperature, a quantity that will be discussed in detail in Chapter 13). The molar
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Fig. 10.8 Room temperature density of Corning Jade glass annealed at three different fictive temperatures, Tf ¼ 700, 735, and 758°C, and rapidly quenched from a hightemperature melt. The vibrational CTE of each glass is found to be 42.3 107/°C, independent of thermal history. The change in room temperature density versus fictive temperature of the annealed glasses gives a configurational CTE of 107 107/°C. The CTE of the liquid can be calculated as the sum of the vibrational and configurational contributions (¼149.3 107/°C). This value is in excellent agreement with the directly measured value of 146.5 107/°C for the liquid. (Modified from M. Potuzak, J.C. Mauro, T.J. Kiczenski, A.J. Ellison, D.C. Allan, J. Chem. Phys. 133 (2010) 091102.)
volume of these samples is measured at room temperature, and the resulting configurational CTE is calculated by [14] 1 ∂V : (10.6) αconf ¼ 3V ∂Tf T , P
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Note that here, the fictive temperature (Tf), described in Chapter 2, replaces the thermal temperature (T) as compared to Eq. (10.1).
10.7 Thermal stresses and thermal shock resistance As stated in the Introduction, the development of a nonuniform temperature distribution in a glass (or any solid) results in the development of strains and stresses. The constitutive equations of the type in Eq. (8.3) governing the relation of the normal component of the strains to the normal components of stresses must be modified as 1 (10.7) σ x ν σ y + σ z + αT E In the simple case of a bar which is restrained in the x-direction and is free to move in the y- and z-directions, the solution of the modified constitutive equations (see Section 8.1) leads to only one nonvanishing stress component: ex ¼
σ x ¼ EαT ,
(10.8)
ez ¼ ey ¼ αT ð1 + νÞ:
(10.9)
and the nonvanishing strains: The normal stress components in an infinite plate (z ¼ thickness direction) cooled such that temperature is a function only of z, are given by E αðT Ta Þ, 1ν where Ta is the average temperature and σx ¼ σy ¼
σ z ¼ 0:
(10.10)
(10.11)
Application of Eqs. (10.10) and (10.11) means the development of a biaxial tension (positive sign of the stress) in glass wherever the temperature is lower than the average and a compression (negative sign of the stress) wherever the situation is opposite. The biaxial stress distribution has the same mathematical form as that of the temperature distribution. It follows that if T were of a parabolic form with outside cooler than the inside (temperature profile produced by normal cooling means), then the stress profile would also be parabolic with tension on the outside and compression on the inside (Fig. 10.9). It may be readily shown that if Ti Ts ¼ Θ, where Ti and Ts are the inside and the surface temperatures, then (Ti Ta) ¼ Θ/3, and (Ta Ts) ¼ 2Θ/3. Hence, for the interior:
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Fig. 10.9 Stress production in glass due to thermal shock.
and for the surface
E Θ σx ¼ σy ¼ α 1ν 3
(10.12)
E 2Θ α σx ¼ σy ¼ 1ν 3
(10.13)
It may also be shown that, for a plate cooled symmetrically at a rate of Φ °C/s, the planar stresses are given by ρCp E (10.14) ΦL 2 α σx ¼ σy ¼ K 3ð1 νÞ where L is the half-thickness of the plate, and ρ, Cp, and K are the density, specific heat, and thermal conductivity of the glass, respectively. Note that, in Eq. (10.14), Φ is positive for cooling and negative for heating. Surface and interior stresses for several other standard geometries have been tabulated by Kingery et al. [15] The outside surface tension developed on cooling can lead to glass fracture. Exposing a glass product to a sudden change in temperature and observing whether the glass fails or not is the basis of the “thermal shock” test. Typically, a number of glass specimens are heated in an oven, and then plunged suddenly into an ice-water mixture. The survivors are put back in the oven at a higher temperature and plunged back into the cold bath. The process is carried until all the samples have failed. A plot of the number of survivors versus the interval of temperature change ΔT (¼oven temperature minus the ice-water temperature) readily yields that value of the interval at which the probability of survival is 50%. This temperature interval is
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termed as the “thermal shock resistance” or the “thermal endurance” of the glass. For glass articles of similar dimensions but of different composition of glasses, the thermal endurance is estimated by sffiffiffiffiffiffiffiffi S K ΔT ¼ m (10.15) αE ρCp where S is the tensile strength of glass and m is the constant [16]. The properties E and (K/ρCp) are relatively insensitive to the composition, and hence the thermal endurance is approximately inversely proportional to the thermal expansion coefficient. For articles of the same composition and shape, the variability clearly arises from the variability of the tensile strength of individual specimens which, in turn, reflects the magnitude of permanent stresses and surface damage due to the manufacturing process. For many situations where the length dimension varies, it has been found that the thermal endurance is approximately given by the relation nS ΔT ¼ pffiffiffi α L
(10.16)
where n is another constant. Thus, in addition to the inverse proportionality to pffiffiffithe thermal expansion coefficient, the thermal endurance decreases as L , that is, thinner glass articles can withstand higher temperature shock. Since the stresses in the interior of the glass are of the opposite sign, it is possible to test the endurance of the interior by “upshocking” of the temperature (for instance, sudden immersion in hot water or in molten salt). The chief problem in carrying out thermal shock tests is the rate at which the outside surface temperature is changed. This, of course, depends on how fast the article is transferred from one bath to the other and the rate of heat transfer between the glass and the immersion bath. It should be recognized that different liquids do not give the same quenching effect (for instance, immersion in liquid N2 is not the most effective way of quenching because of the formation of a vapor cushion). Likewise, immersion from a 100°C oven to 4°C water does not impart the same thermal shock as immersion from a 160°C oven to 64°C water. Another problem is the effect of the immersion fluid on the strength of glass. Water generally has a corrosive effect compared to oils (see Chapter 17). Hence, immersion from a 160°C oven to 64°C water will not give the same thermal endurance results as immersion from a 160°C oven to 64°C oils. Despite these experimental problems, thermal shock resistance tests are widely accepted quality control procedures for accelerated life testing in severe environments.
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Fig. 10.10 Harold Rawson (1927–2001), pioneer in glass sealing technologies.
10.8 Glass greats: Harold Rawson Harold Rawson (1927–2001, Fig. 10.10) grew up in Rochdale, England, and graduated with a degree in Glass Technology from the University of Sheffield in 1947. Following graduation, Rawson joined the Glass Research Group of the British Thomson-Houston Company, where he developed new glass sealing technologies for the emerging semiconductor electronics industry. In 1964, Rawson joined the University of Sheffield as a lecturer in the Department of Glass Technology, where he also earned his doctoral degree in 1973. In 1975, he became the W.E.S. Turner Professor of Glass Technology at Sheffield. While at Sheffield, Rawson made the original prediction that the optical attenuation of glass could be brought below 20 dB/ km by reducing impurities, providing encouragement to Charles Kao for his pioneering work in optical fiber in the mid-1960s [17], which eventually led to the 2009 Nobel Prize in Physics for Kao. Rawson was well known for both his excellent research and teaching abilities. He was the author of three highly regarded books, including the classic Inorganic Glass Forming Systems, published in 1967. Rawson became an honorary fellow of the Society of Glass Technology in 1997 and was a recipient of the President’s Award from the International Commission on Glass in 1998.
Summary The thermal expansion of a glass results from the thermal energy-induced change in the time-averaged separation between atoms vibrating in an
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asymmetric potential well. Thermal expansion coefficient is measured as the change in length per unit rise in temperature. It is commonly measured using a pushrod dilatometer. On heating a glass, the specimen expands much like a solid till the onset of glass transition. During transition, the glass may contract depending on its prior cooling history. Rapid expansion, typical of a liquid, occurs thereafter. Thermal expansion coefficient of a glass increases with the alkali content. A typical value for a common commercial soda lime silicate glass is 90 107/°C, whereas that of sodium borosilicates varies between 30–60 107/°C, and of silica glass is 5 107/°C. The thermal expansion coefficient of a molten liquid includes contributions from both anharmonic vibrations and configurational transitions. This is in contrast to the glassy state, where the thermal expansion is entirely due to vibrations. When hot glass products are quenched, the resulting tensile stress in the surface could result in glass fracture. Thermal shock resistance or thermal endurance is inversely proportional to the thermal expansion coefficient and square root of the glass thickness. Thus, silica glass is quite resistant to thermal shock relative to the soda limes.
Online Resources (1) “Glass-ceramic versus tempered glass in BBQ Grills” by SCHOTT: https://www.youtube.com/watch?v¼-HvxdOt4YUA. (2) “Glass Bakeware that Shatters” by consumer reports: https://www. youtube.com/watch?v¼UyhdMa1ikKM.
Exercises (1) Describe how you would measure the density of a 15Na2O10CaO75SiO2 (mol%) glass from 150°C to 400°C. (2) Some “bead” seals (cylindrical glass sealed over a round metal wire) are notorious for having excessive hoop stresses in the glass near the metal interface. What thermal shock procedure would you recommend to assure the quality of the seals? (3) A semiinfinite glass plate with a thickness of 2L and expansion coefficient α has been cooled such that the temperature profile through the thickness is parabolic (the temperature is the highest in the middle).
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Develop an expression for the stress profile. Show that the depth of the tension in the surface is 0.216 of the total plate thickness. (4) Write typical values of αm for a common soda lime container glass measured using a pushrod dilatometer when (i) heated from 20°C to 300°C, (ii) cooled from 300°C to 20°C, (iii) heated from 20°C to 550°C, and (iv) cooled from 550°C to 20°C. (5) How are the above measurements affected when the glass has bubbles covering about 10% of the cross-sectional area? (6) It is proposed to modify the composition of a 15Na2O10CaO75SiO2 (mol%) glass such that it would have a slightly higher annealing point, but slightly lower thermal expansion coefficient. What “minor” composition change would you recommend? (7) Under what conditions a specimen of soda lime silica window glass display a negative thermal expansion coefficient regardless of the measurement technique? (8) A composite glass fiber of diameter d is made by fusing two equal semicircular cross-section area strips together, “beading” them in the middle, and then gently drawing the bead. The fiber solidifies at the lower of the two values of Tg and generally assumes a curved shape. Assume that the elastic properties of the two glasses are identical. Develop a simplified formula to calculate Δα between the two glasses by measuring the mid-point deflection h and cord length L of the segment formed by the curved fiber. (Hint: Use ts ¼ tg ¼ d/2; Es ¼ Eg; and νs ¼ νg in Eq. (10.4).) [Ans: Δα ¼ 16hd/[3ΔT(L2 + 4h2)], where ΔT is the difference from the lower Tg to room temperature. See also A. Roth, Vacuum Sealing Techniques, Pergamon Press, Oxford, 1966, p. 109]. (9) Why do you think the thermal expansion coefficient of fused silica is nearly zero? (10) What glass and metal seal combinations would you recommend for the following lamp applications? (i) Household incandescent lamp, where the maximum service temperature requirement is a sustained electrical insulation between the current leads at about 150°C, and there is little concern for thermal shock. (Ans: Although the lamp envelope is made from the most inexpensive soda lime silicate glass, the mount structure is actually made using a potash soda lead glass which has a longer working range and a much higher electrical resistivity than the soda lime silica glass. The seal is made over “Dumet” wire, which is a copper clad 42Ni-Fe alloy.
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Because of it being a composite wire, the Dumet wire has an anisotropic thermal expansion coefficient. The expansion of the glass is “matched” well for the radial expansion of the alloy). (ii) Photoflash lamp, where the envelope experiences a sudden temperature, but the outer wall temperature does not exceed 200°C. Because of the temperature pulse, there is some need for superior thermal shock resistance relative to that of the soft glasses. (Ans: These lamps are made from sodium borosilicate glasses having a CTE of 53 10–7/°C. The seal is made using Kovar alloy.) (iii) Glass-halogen lamp where the service temperature requirement is a sustained 600°C (hence, the glass must also have high electrical resistivity at 600°C). (Ans: These lamps are made using alkaline earth aluminosilicate glasses. Because of the absence of alkalis, they have very high resistivities. The seal is made over Mo metal.)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
R.N. Work, J. Res. Natl. Bur. Stand. 47 (1951) 80. W.A. Plummer, H.E. Hagy, Appl. Opt. 7 (1968) 825. H.E. Hagy, J. Am. Ceram. Soc. 62 (1979) 60. A.K. Varshneya, M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mater. Sci. and Technol, vol. 22, Academic Press, 1982, , pp. 241–305. A.K. Varshneya, J. Am. Ceram. Soc. 63 (1980) 311. W.C. Young, Roark’s Formulas for Stress and Strain, 6th ed., McGraw-Hill Book Co., New York, 1989 Table 24, Case 15a, pp. 432, 446–448. J.F. Jongste, F.E. Prins, G.C.A.M. Janssen, S. Radelaar, Proc. Mat. Res. Soc. Symp. 130 (1989) 321. G.G. Stoney, Proc. R. Soc. 82 (1909) 172. R.B. Sosman, The Properties of Silica, Amer. Chem. Soc., New York, 1927, 367 D.F. Gibbons, Phys. Chem. Solids 11 (1959) 246. M.R. Vukcevich, J. Non-Cryst. Solids 11 (1972) 25. H.A. Miska, Ceramics and Glasses, Engineered Materials Handbook, vol. 4, ASM Publ., 1992, pp. 1016–1020 T. Lakatos, L.-G. Johansson, B. Simminskold, Glastek. Tidsk. 28 (1973) 69. M. Potuzak, J.C. Mauro, T.J. Kiczenski, A.J. Ellison, D.C. Allan, J. Chem. Phys. 133 (2010) 091102. W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, second ed., John Wiley, New York, 1975, 819. F.V. Tooley, Handbook of Glass Manufacture, vol. 1, Ogden Publ., New York, 1953, 18 C. DeCusatis (Ed.), Handbook of Fiber Optic Data Communication, third ed., Elsevier, 2008, p. 9.
Thermal expansion of glass 10.1 Introduction When a glass is heated, it generally expands. If the temperature over the body of glass is equal everywhere and the body is not restrained, then there will be no development of stress in the body. If, on the other hand, there were nonuniform heating over the body, then the different layers of glass would attempt to expand differently. The enforcement of the compatibility criteria (see Appendix) results in a different set of dimensional changes, and consequently, stresses develop. The tensile stress components could lead to glass failure (which is discussed in detail in Chapter 18). The magnitude of the stresses so generated is related to thermal expansion. The expansion due to heat flow in or the contraction due to heat flow out of the body is, therefore, an important consideration in product design. Thus, architectural glass window panes need to tolerate a temperature gradient across their thickness created by the difference between room temperature and the temperature on the outside. A hot glass-baking dish should be able to withstand the “thermal shock” when set on a cold kitchen countertop. It may be realized that thermal compatibility principles apply not only to the various parts of the glass body but also to other materials which have been joined to the glass. Thus, the success of glass-to-metal seals in a variety of today’s products critically hinges on the thermal expansion of glass (and of the metal). Consideration of thermal expansion is also important in the annealing of glassware. Later in Chapter 13, we will show that annealing refers to the release of thermal stresses generated from the nonuniform cooling of glass through the glass transition range during glass forming. In this chapter, we will review the thermal expansion property of the glasses.
10.2 Definitions Over the temperature range T1 to T2, the mean or the average volume expansion coefficient βm is given by Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00010-9
© 2019 Elsevier Inc. All rights reserved.
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βm ¼
V2 V1 V1 ðT2 T1 Þ
(10.1)
where Vi is the volume at temperature Ti. The change in volume is to be measured under closed isobaric (constant pressure) conditions. The true or instantaneous volume expansion coefficient β is given by 1 ∂V β¼ : (10.2) V ∂T P The corresponding linear expansion coefficients (α) are obtained by replacing volume V by length L. Appropriate units for the coefficient of thermal expansion (CTE) are /°C in the cgs system and K1 in the SI system. Also, for small expansion coefficients, β ¼ 3α:
(10.3)
10.3 Methods of thermal expansion measurement In most practical applications linear thermal expansion is of primary interest. It is generally measured by three methods: pushrod dilatometers, interferometry, and standard glass seals. The experimental setup for a typical commercial pushrod dilatometer is shown in Fig. 10.1. The specimen, held against the flat wall of the measuring system, pushes on a horizontal spring-loaded pushrod on heating. This change relative to either the system tube itself or to a second pushrod against a standard specimen is measured using an LVDT (linear variable differential transducer). The measuring system and the pushrods are generally made out of alumina or fused silica. An equal length platinum sample is generally a good standard to use. The key to getting good data is the uniformity of temperature over the specimen and the standard; both of which should have flat, parallel ends. The differential accuracy improves if the unknown and the
Fig. 10.1 Schematics of a double pushrod differential dilatometer. (Courtesy: Theta Industries, Inc., Port Washington, New York.)
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standard are close in expansion characteristics. Some commercial differential pushrod dilatometers can be accurate to as little as 0.5% which is generally quite adequate. A typical plot from a dilatometer (after adding the expansion of the standard specimen) is shown in Fig. 10.2. Note the rapid rise in expansion which marks the onset of the glass transition range. The curved hook downwards is an artifact of the measuring technique itself. At this temperature, the expansion of the specimen is countered by an apparent contraction due to specimen softening (pushrods begin to penetrate into the specimen or the latter begins to sag because the viscosity is about 109–1010 Pa s). This temperature is called the dilatometric softening temperature Td. The measurements should be stopped soon after. (Failure to stop the furnace from heating could result in the specimen getting much too soft and possible damage to the system due to glass adherence.) The return (contraction) path is never parallel to the expansion path in the glass transition range. This is explained further in Chapter 13. For now, it suffices to suggest that, for many applications, it is the contraction curve which is important in the consideration of stress generation and not the expansion
Fig. 10.2 Effect of the cooling rate history on the expansion of a glass.
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curve. For sealing with glass, the thermal contraction data should be computed from the set point to the room temperature. The set point may be thought to be a cooling rate-dependent idealized temperature below which the glass may be considered an elastic solid. In the interferometric method, three carefully ground and polished pieces, two of a known and one of the unknown glass, separate two optically flat and parallel surfaces. As the temperature increases, the separation also increases, which can be monitored by interferometric methods [1, 2]. When a glass is to be employed for sealing purposes, a more accurate method of determining the expansion is by employing the unknown as a component in a standard seal. Convenient seal forms are the sandwich seal (described in ASTM F144-73) and the trident seal [3]. A sandwich seal is formed by fusing three plates to each other in such a manner that the outer pieces may be a known glass, and the inner piece is the unknown glass. A trident seal is a variation of the sandwich seal. Here the pieces neither have to be plates nor have to be fused to each other along the entire length. They merely have to be joined together at the ends. One then has to make a measurement of the optical retardation (generally at the center) and convert this optical retardation to a stress (see “Photoelastic Properties,” Section 19.2.6). Once the stress is known, the mismatch is calculated [4] from standard formulas. A thermal contraction coefficient may be extracted [5] from the mismatch if the temperature interval ΔT between the set point and the room temperature is known. To examine the suitability of a thick or thin glass film fired onto a ceramic, metal, or glass substrate, it is often useful to measure the deflection of composite strips or circular plates. In particular, the depression h in a circular disk of radius r, thickness ts, over which a glass coating of thickness tg has been applied can be converted to yield the differential thermal contraction coefficient Δα using the expression [6] Δα ¼
ts2 K1p h , 3ΔT tg + ts r 2
(10.4)
where K1p ¼ 4 + 6
2 3 tg tg Eg tg 1 νg 1 νs Es ts +4 + + ts ts Es 1 νg ts Eg 1 νs tg
Here, E and ν are the Young’s modulus and Poisson’s ratio; the subscripts s and g refer to the substrate and the glass coating. For glass coating thickness tg much less than the substrate thickness ts, the above formula reduces to
Thermal expansion of glass
Δα ¼
ht 2s 3r 2 ΔTt g
1 νg Es , Eg 1 νs
257
(10.5)
which is also obtained [7] by extending Stoney’s formula [8] for stresses in a bimetallic disk. A semiquantitative technique used by lampworkers to determine compatibility of sealing between two glasses is the composite fiber method discussed in Exercise 8.
10.4 Thermal expansion versus composition and temperature Before studying various thermal expansion data, it should be noted that most expansion coefficients in the glass literature are quoted in terms of 10–7/°C. On the other hand, metallurgists quote the metal expansion data at 10–6/°C or ppm. (This, of course, is important during a conversation.) Yet another issue, often, is the range of temperatures over which the value of the expansion coefficient is quoted. Glass manufacturers generally quote the mean expansion coefficient over 20–300°C, which is the assumed range for most of the expansion coefficient values cited unless specified otherwise. Of all the single-component glasses, silica glass has perhaps the lowest thermal expansion coefficient. The average value between 20°C and 1000°C is roughly 5.5 10–7/°C. (Because of this rather low value of α, silica glass is preferred for products which require a high thermal shock resistance; see below.) In marked contrast to the expansion of silica glass, the mean linear expansion coefficient from 20°C for cristobalite increases from about 55 10–7/°C to about 200 10–7/°C at 200°C, and it undergoes a large expansion increase at the α ! β conversion to an effective mean expansion coefficient of 550 10–7/°C at about 270°C, subsequently decreasing to that of silica glass at temperatures above 1000°C. (The mean expansion coefficient values stated here are one-third of the measured mean volumetric expansion coefficients, see Ref. [9]). The contrast between the expansions of silica with other glass-forming oxides such as B2O3 (150 10–7/°C) and GeO2 (77 10–7/°C) is also noticeable. In the vicinity of 0 K, vitreous silica and several other tetrahedral network formers display negative expansion (Fig. 10.3). For vitreous silica, the anomaly continues [10] from 0 K to about room temperature. This anomalous behavior, similar to that for the elastic properties of silica has been explained by Vukcevich [11] in terms of changing population of SiOSi bond angle between
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Fig. 10.3 Thermal expansion curves of some single-component glasses at low temperatures. (Modified from J.T. Krause, C.R. Kurkjian, J. Am. Ceram. Soc. 51 (1968) 226.)
145° and 138° (Section 8.2.4). More on the anomalous behavior of silica at low temperatures is discussed in the following chapter on heat capacity. The addition of alkali to the fully connected silica network breaks up the oxygen bridges, resulting in a monotonically increasing thermal expansion coefficient as a function of the alkali content (Fig. 10.4A): the higher the alkali content, the larger the expansion (Fig. 10.4B). In the alkali borate glasses, however, pronounced minima in the thermal expansion coefficient are observed as a function of the alkali concentration (Fig. 10.5). This
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Fig. 10.4 (A) Thermal expansion coefficients of R2OSiO2 (R ¼ Li, Na, K) glasses as a function of composition. (B) Thermal expansion curve for xNa2O(100 x)SiO2 glasses.
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Fig. 10.5 Thermal expansion coefficients of R2O-B2O3 glasses as a function of composition.
observation, otherwise called the boron anomaly, has already been discussed in Section 5.6. B2O3 addition to SiO2 also increase the expansion coefficient monotonically (Fig. 5.18) despite the absence of nonbridging oxygens. Here, the expansion increases as a result of the freedom allowed by the triangular boron connections. The addition of TiO2 to silica lowers [12] the expansion coefficient. As seen in Fig. 10.6, a 7.5TiO292.5SiO2 (wt%) glass made by the chemical vapor deposition (CVD) process (Chapter 22) has an expansion coefficient of 0 0.3 10–7/°C over the 5–35°C range (Corning “Ultra Low Expansion” ULE glass). Such zero expansion glasses are used as honeycomb substrates for space-based lightweight mirrors. Table 10.1 lists the expansion coefficients of some commercial glasses. The soda lime silicate glasses generally have an expansion coefficient of 85–95 10–7/°C. Between the soda lime silicate glasses on one side and vitreous silica on the other side are the alkali borosilicate and the alkaline earth aluminosilicate glasses. The expansions of the alkali borosilicates range between 10 10–7/°C (Vycor) and about 55 10–7/°C (GE 706, Corning 7052), and those of the alkaline earth aluminosilicates around 45 10–7/°C (GE180).
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Fig. 10.6 Thermal expansion of TiO2-SiO2 glasses as a function of temperature. (Modified from Ref. H.A. Miska, in: Ceramics and Glasses, Engineered Materials Handbook, vol. 4, ASM Publ., 1992, pp. 1016–1020.)
Table 10.1 Thermal expansion coefficients of some commercial glasses Expansion coefficient 1027/°C Glass code Type 0–300°C Set point — 25°C
GE 001 GE 008 GE 012 Corning 1720 GE 706 GE 725 Corning 7720 Corning 7740 Corning 7913 Corning 7940
Potash soda lead Soda lime Potash soda lead Aluminosilicate Borosilicate Borosilicate Borosilicate Borosilicate 96% silica Fused silica
93.5 93.5 89.5 42 48 35.5 36 32.5 7.5 5.5
101 105 97 52 55 39 43 35 5.5 3.5
Additivity factors for the various components in the glasses have been obtained by various workers. Some of these are listed in Table 10.2. Factors provided by Lakatos et al. [13] apply closely to the commercial soda lime silicate compositions.
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Table 10.2 Additivity factor, pi, for calculating thermal expansion coefficientsa Oxide Winkelmann and Schottb English and Turnerc Gilard and Dubruld
SiO2 B2 O 3 Na2O K2O MgO CaO ZnO BaO PbO Al2O3
0.267 0.033 3.333 2.833 0.033 1.667 0.60 1.0 1.0 1.667
0.050 0.653 4.16 3.90 0.45 1.63 0.70 1.40 1.06 0.14
0.04 0.4 + 0.01w 5.1 0.0333w 4.2 0.0333w 0 0.75 + 0.035w 0.775 0.025w 0.91 + 0.014w 1.15 0.005w 0.2
P α ¼ piwi 107/°C; wi ¼ wt%. A. Winkelmann, O. Schott, Ann. Phys. (Leipzig) 51 (1984) 735. c S. English, W.E.S. Turner, J. Am. Ceram. Soc. 12 (1929) 760. d P. Gilard, L. Dubrul, Verre Silic. Ind. 5 (1934) 122. a
b
10.5 Concepts of glass expansion In ionically bonded crystalline materials, one thinks of the thermal expansion to arise from the asymmetric shape of the Condon-Morse potential well diagram. The asymmetric shape arises because of the anharmonic terms in the expression for the potential well. When the energy is raised by adding thermal energy, the ion is able to vibrate between points B and C (Fig. 10.7). The average location of the ion appears to be A0 , A00 , etc., and hence, the effective separation between the ions increases, as shown in Fig. 10.7. (The reader should note that the dissociation or sublimation energy of ions is related to the depth of the potential well, the Young’s modulus is related to the curvature of the trough, and the thermal expansion is related to the asymmetry of the potential function.) However, much like the argument presented for the elasticity in glass (Chapter 8), network structures can also expand or contract by a bondbending mechanism. Thus, in effect, the expansion of a glass may have very little resemblance to the thermal expansion of a crystal of identical composition.
10.6 Configurational versus vibrational contributions to thermal expansion The CTE of a glass-forming liquid is generally higher than that of its corresponding glass. This is because the CTE of the liquid includes contributions
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Fig. 10.7 Change in the atomic separation, r, with temperature. At 0 K, the equilibrium separation is r0. Thermal energy at T1 causes oscillations between ra and rb; the mean position re is greater than r0 because of the asymmetry of the potential function.
from both configurational and vibrational motions, whereas in the glassy state the configurational transitions are frozen so that the CTE is entirely a result of anharmonic vibrations within a single energy well. In contrast, the liquid state is highly fluid so the sample can transition between different atomic configurations at different temperatures. These different atomic configurations generally correspond to different molar volumes, and it is this change in volume that gives rise to the configurational contribution of a liquid’s thermal expansion coefficient. Hence, in a glass the CTE is equal to the vibrational contribution only, that is, αglass ¼ αvib, whereas in a liquid the CTE is the sum of vibrational and configurational contributions: αliquid ¼ αvib + αconf ¼ αglass + αconf. An example of separating the CTE of the liquid into vibrational and configurational contributions is shown in Fig. 10.8. Here the vibrational CTE (¼αglass) is obtained through standard dilatometry of a glass sample. The configurational CTE is obtained by equilibrating glass samples in a furnace at different temperatures and then rapidly quenching to room temperature to freeze in the particular configurations (i.e., to freeze in particular values of the fictive temperature, a quantity that will be discussed in detail in Chapter 13). The molar
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Fig. 10.8 Room temperature density of Corning Jade glass annealed at three different fictive temperatures, Tf ¼ 700, 735, and 758°C, and rapidly quenched from a hightemperature melt. The vibrational CTE of each glass is found to be 42.3 107/°C, independent of thermal history. The change in room temperature density versus fictive temperature of the annealed glasses gives a configurational CTE of 107 107/°C. The CTE of the liquid can be calculated as the sum of the vibrational and configurational contributions (¼149.3 107/°C). This value is in excellent agreement with the directly measured value of 146.5 107/°C for the liquid. (Modified from M. Potuzak, J.C. Mauro, T.J. Kiczenski, A.J. Ellison, D.C. Allan, J. Chem. Phys. 133 (2010) 091102.)
volume of these samples is measured at room temperature, and the resulting configurational CTE is calculated by [14] 1 ∂V : (10.6) αconf ¼ 3V ∂Tf T , P
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Note that here, the fictive temperature (Tf), described in Chapter 2, replaces the thermal temperature (T) as compared to Eq. (10.1).
10.7 Thermal stresses and thermal shock resistance As stated in the Introduction, the development of a nonuniform temperature distribution in a glass (or any solid) results in the development of strains and stresses. The constitutive equations of the type in Eq. (8.3) governing the relation of the normal component of the strains to the normal components of stresses must be modified as 1 (10.7) σ x ν σ y + σ z + αT E In the simple case of a bar which is restrained in the x-direction and is free to move in the y- and z-directions, the solution of the modified constitutive equations (see Section 8.1) leads to only one nonvanishing stress component: ex ¼
σ x ¼ EαT ,
(10.8)
ez ¼ ey ¼ αT ð1 + νÞ:
(10.9)
and the nonvanishing strains: The normal stress components in an infinite plate (z ¼ thickness direction) cooled such that temperature is a function only of z, are given by E αðT Ta Þ, 1ν where Ta is the average temperature and σx ¼ σy ¼
σ z ¼ 0:
(10.10)
(10.11)
Application of Eqs. (10.10) and (10.11) means the development of a biaxial tension (positive sign of the stress) in glass wherever the temperature is lower than the average and a compression (negative sign of the stress) wherever the situation is opposite. The biaxial stress distribution has the same mathematical form as that of the temperature distribution. It follows that if T were of a parabolic form with outside cooler than the inside (temperature profile produced by normal cooling means), then the stress profile would also be parabolic with tension on the outside and compression on the inside (Fig. 10.9). It may be readily shown that if Ti Ts ¼ Θ, where Ti and Ts are the inside and the surface temperatures, then (Ti Ta) ¼ Θ/3, and (Ta Ts) ¼ 2Θ/3. Hence, for the interior:
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Fig. 10.9 Stress production in glass due to thermal shock.
and for the surface
E Θ σx ¼ σy ¼ α 1ν 3
(10.12)
E 2Θ α σx ¼ σy ¼ 1ν 3
(10.13)
It may also be shown that, for a plate cooled symmetrically at a rate of Φ °C/s, the planar stresses are given by ρCp E (10.14) ΦL 2 α σx ¼ σy ¼ K 3ð1 νÞ where L is the half-thickness of the plate, and ρ, Cp, and K are the density, specific heat, and thermal conductivity of the glass, respectively. Note that, in Eq. (10.14), Φ is positive for cooling and negative for heating. Surface and interior stresses for several other standard geometries have been tabulated by Kingery et al. [15] The outside surface tension developed on cooling can lead to glass fracture. Exposing a glass product to a sudden change in temperature and observing whether the glass fails or not is the basis of the “thermal shock” test. Typically, a number of glass specimens are heated in an oven, and then plunged suddenly into an ice-water mixture. The survivors are put back in the oven at a higher temperature and plunged back into the cold bath. The process is carried until all the samples have failed. A plot of the number of survivors versus the interval of temperature change ΔT (¼oven temperature minus the ice-water temperature) readily yields that value of the interval at which the probability of survival is 50%. This temperature interval is
Thermal expansion of glass
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termed as the “thermal shock resistance” or the “thermal endurance” of the glass. For glass articles of similar dimensions but of different composition of glasses, the thermal endurance is estimated by sffiffiffiffiffiffiffiffi S K ΔT ¼ m (10.15) αE ρCp where S is the tensile strength of glass and m is the constant [16]. The properties E and (K/ρCp) are relatively insensitive to the composition, and hence the thermal endurance is approximately inversely proportional to the thermal expansion coefficient. For articles of the same composition and shape, the variability clearly arises from the variability of the tensile strength of individual specimens which, in turn, reflects the magnitude of permanent stresses and surface damage due to the manufacturing process. For many situations where the length dimension varies, it has been found that the thermal endurance is approximately given by the relation nS ΔT ¼ pffiffiffi α L
(10.16)
where n is another constant. Thus, in addition to the inverse proportionality to pffiffiffithe thermal expansion coefficient, the thermal endurance decreases as L , that is, thinner glass articles can withstand higher temperature shock. Since the stresses in the interior of the glass are of the opposite sign, it is possible to test the endurance of the interior by “upshocking” of the temperature (for instance, sudden immersion in hot water or in molten salt). The chief problem in carrying out thermal shock tests is the rate at which the outside surface temperature is changed. This, of course, depends on how fast the article is transferred from one bath to the other and the rate of heat transfer between the glass and the immersion bath. It should be recognized that different liquids do not give the same quenching effect (for instance, immersion in liquid N2 is not the most effective way of quenching because of the formation of a vapor cushion). Likewise, immersion from a 100°C oven to 4°C water does not impart the same thermal shock as immersion from a 160°C oven to 64°C water. Another problem is the effect of the immersion fluid on the strength of glass. Water generally has a corrosive effect compared to oils (see Chapter 17). Hence, immersion from a 160°C oven to 64°C water will not give the same thermal endurance results as immersion from a 160°C oven to 64°C oils. Despite these experimental problems, thermal shock resistance tests are widely accepted quality control procedures for accelerated life testing in severe environments.
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Fig. 10.10 Harold Rawson (1927–2001), pioneer in glass sealing technologies.
10.8 Glass greats: Harold Rawson Harold Rawson (1927–2001, Fig. 10.10) grew up in Rochdale, England, and graduated with a degree in Glass Technology from the University of Sheffield in 1947. Following graduation, Rawson joined the Glass Research Group of the British Thomson-Houston Company, where he developed new glass sealing technologies for the emerging semiconductor electronics industry. In 1964, Rawson joined the University of Sheffield as a lecturer in the Department of Glass Technology, where he also earned his doctoral degree in 1973. In 1975, he became the W.E.S. Turner Professor of Glass Technology at Sheffield. While at Sheffield, Rawson made the original prediction that the optical attenuation of glass could be brought below 20 dB/ km by reducing impurities, providing encouragement to Charles Kao for his pioneering work in optical fiber in the mid-1960s [17], which eventually led to the 2009 Nobel Prize in Physics for Kao. Rawson was well known for both his excellent research and teaching abilities. He was the author of three highly regarded books, including the classic Inorganic Glass Forming Systems, published in 1967. Rawson became an honorary fellow of the Society of Glass Technology in 1997 and was a recipient of the President’s Award from the International Commission on Glass in 1998.
Summary The thermal expansion of a glass results from the thermal energy-induced change in the time-averaged separation between atoms vibrating in an
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269
asymmetric potential well. Thermal expansion coefficient is measured as the change in length per unit rise in temperature. It is commonly measured using a pushrod dilatometer. On heating a glass, the specimen expands much like a solid till the onset of glass transition. During transition, the glass may contract depending on its prior cooling history. Rapid expansion, typical of a liquid, occurs thereafter. Thermal expansion coefficient of a glass increases with the alkali content. A typical value for a common commercial soda lime silicate glass is 90 107/°C, whereas that of sodium borosilicates varies between 30–60 107/°C, and of silica glass is 5 107/°C. The thermal expansion coefficient of a molten liquid includes contributions from both anharmonic vibrations and configurational transitions. This is in contrast to the glassy state, where the thermal expansion is entirely due to vibrations. When hot glass products are quenched, the resulting tensile stress in the surface could result in glass fracture. Thermal shock resistance or thermal endurance is inversely proportional to the thermal expansion coefficient and square root of the glass thickness. Thus, silica glass is quite resistant to thermal shock relative to the soda limes.
Online Resources (1) “Glass-ceramic versus tempered glass in BBQ Grills” by SCHOTT: https://www.youtube.com/watch?v¼-HvxdOt4YUA. (2) “Glass Bakeware that Shatters” by consumer reports: https://www. youtube.com/watch?v¼UyhdMa1ikKM.
Exercises (1) Describe how you would measure the density of a 15Na2O10CaO75SiO2 (mol%) glass from 150°C to 400°C. (2) Some “bead” seals (cylindrical glass sealed over a round metal wire) are notorious for having excessive hoop stresses in the glass near the metal interface. What thermal shock procedure would you recommend to assure the quality of the seals? (3) A semiinfinite glass plate with a thickness of 2L and expansion coefficient α has been cooled such that the temperature profile through the thickness is parabolic (the temperature is the highest in the middle).
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Develop an expression for the stress profile. Show that the depth of the tension in the surface is 0.216 of the total plate thickness. (4) Write typical values of αm for a common soda lime container glass measured using a pushrod dilatometer when (i) heated from 20°C to 300°C, (ii) cooled from 300°C to 20°C, (iii) heated from 20°C to 550°C, and (iv) cooled from 550°C to 20°C. (5) How are the above measurements affected when the glass has bubbles covering about 10% of the cross-sectional area? (6) It is proposed to modify the composition of a 15Na2O10CaO75SiO2 (mol%) glass such that it would have a slightly higher annealing point, but slightly lower thermal expansion coefficient. What “minor” composition change would you recommend? (7) Under what conditions a specimen of soda lime silica window glass display a negative thermal expansion coefficient regardless of the measurement technique? (8) A composite glass fiber of diameter d is made by fusing two equal semicircular cross-section area strips together, “beading” them in the middle, and then gently drawing the bead. The fiber solidifies at the lower of the two values of Tg and generally assumes a curved shape. Assume that the elastic properties of the two glasses are identical. Develop a simplified formula to calculate Δα between the two glasses by measuring the mid-point deflection h and cord length L of the segment formed by the curved fiber. (Hint: Use ts ¼ tg ¼ d/2; Es ¼ Eg; and νs ¼ νg in Eq. (10.4).) [Ans: Δα ¼ 16hd/[3ΔT(L2 + 4h2)], where ΔT is the difference from the lower Tg to room temperature. See also A. Roth, Vacuum Sealing Techniques, Pergamon Press, Oxford, 1966, p. 109]. (9) Why do you think the thermal expansion coefficient of fused silica is nearly zero? (10) What glass and metal seal combinations would you recommend for the following lamp applications? (i) Household incandescent lamp, where the maximum service temperature requirement is a sustained electrical insulation between the current leads at about 150°C, and there is little concern for thermal shock. (Ans: Although the lamp envelope is made from the most inexpensive soda lime silicate glass, the mount structure is actually made using a potash soda lead glass which has a longer working range and a much higher electrical resistivity than the soda lime silica glass. The seal is made over “Dumet” wire, which is a copper clad 42Ni-Fe alloy.
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Because of it being a composite wire, the Dumet wire has an anisotropic thermal expansion coefficient. The expansion of the glass is “matched” well for the radial expansion of the alloy). (ii) Photoflash lamp, where the envelope experiences a sudden temperature, but the outer wall temperature does not exceed 200°C. Because of the temperature pulse, there is some need for superior thermal shock resistance relative to that of the soft glasses. (Ans: These lamps are made from sodium borosilicate glasses having a CTE of 53 10–7/°C. The seal is made using Kovar alloy.) (iii) Glass-halogen lamp where the service temperature requirement is a sustained 600°C (hence, the glass must also have high electrical resistivity at 600°C). (Ans: These lamps are made using alkaline earth aluminosilicate glasses. Because of the absence of alkalis, they have very high resistivities. The seal is made over Mo metal.)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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