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Density and molar volume
CHAPTER 7
Density and molar volume 7.1 Definitions The density of a substance is defined as the mass per unit volume and is an intensive property. Appropriate units are g/cm3 or g/cc (in cgs system) and kg m3 in the SI system. The dependence of density on temperature is given by the volume thermal expansion coefficient. Since glasses may, in general, be regarded as solutions, a more useful property is the molar volume V, defined as the volume of 1 mol of glass. We then proceed to define partial molar volumes of the various structural units constituting the glass. The partial molar volume vi of a species i in a solution is defined by ∂V (7.1) vi ¼ ∂ni nj , T , P and hence, V¼
X
ni vi :
i
In essence, the total molar volume is treated as an extensive property in terms of the partial molar volumes of individual species or structural groups. The partial molar volume information can be extracted from the density data of glasses where the constituents have been systematically varied. If an Na+ ion is replaced by K+ in the glass and occupies the same site (without changing its size), it is clear that, although the glass density would increase because of the higher atomic weight of K+, the partial molar volume of the alkali ion would not change. Increase in the molar volume, if any, would be an increase in the partial molar volume of K+ relative to Na+. This is likely to have some relation to the relative ionic sizes of the two ions. One may readily note that, although the measurement of density changes in a family of glasses may provide only the trends, the extraction of the partial molar volumes from the density data provides further insight into structure while canceling out the effect of atomic masses.
Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00007-9
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7.2 Methods of measurement Density is measured most commonly by weighing a suitably selected piece of glass in air and dividing this weight by the buoyancy (the reduction in weight) when suspended in a fluid such as water. According to Archimedes’ principle, the buoyancy equals the weight of the displaced fluid. Thus, if wa is the weight of the specimen in air, and wb is the weight of the specimen when immersed in the fluid, then the buoyancy is wa wb and the density ρ ¼ ρmwa/(wa wb), where ρm is the density of the fluid (¼1.0 g/cm3 for water). As some glasses may interact with water, a suitable inert fluid such as kerosene may be selected. A kit supplied by scientific balance manufacturers can be used to obtain density measurements accurate to 0.001 g/cm3. Automatic measurements of density can be carried out using pycnometers which utilize He gas as the displacement fluid. A small sample, of the order of a few mg, is all that is needed to make the measurement. The accuracy of such instruments is generally about 0.002 g/cm3. In a manufacturing environment, a comparative density (easily accurate to 0.0002 g/cm3) is more rapidly obtained using a gradient column. The gradient column consists of a vertically held, one-end closed, long (often 1–1.5 m) glass tube containing a liquid with a linear density gradient. To prepare the density gradient, a heavy liquid such as sym-tetrabromoethane (ρ ¼ 2.96 g/cm3) or methylene iodide (ρ ¼ 3.32 g/cm3) is first poured into the tube and then a lighter liquid such as isopropyl salicylate (ρ ¼ 1.1 g/cm3) is gently poured to float above the heavy liquid.1 With time, the liquids diffuse into each other establishing a gradient with the help of the gravity. The gradient is calibrated by allowing density standards to gently float in the column at different heights. The density of an unknown glass is obtained by reading its floatation level against a pre-calibrated scale placed adjacent to the column. Another method also used commonly is the sink-float method, where the unknown specimen is gently dropped in a tube containing a slightly denser solution of organic liquids such as the ones stated above. The temperature around the tube is gradually changed until the previously floating specimen begins to sink. Calibration of the liquid’s density against temperature yields the density of the unknown. (See ASTM standard C729-75.)
1
Many of these liquids may pose health hazards. The reader is advised to check with the suppliers regarding their handling procedures.
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7.3 Dependence on cooling rate, temperature, and composition As stated in Section 2.2, the volume variation upon cooling is fundamental to the transformation of a supercooled liquid into a glass. Except for a few anomalous cases, such as silica, a fast-cooled glass generally has a higher volume and, hence, lower density, relative to a slow-cooled glass of the same composition. Because density is inversely proportional to the volume, the fractional change in density with a ΔT change in temperature is given by 3αΔT, where α is the linear thermal expansion coefficient. Later, in Chapters 10 and 13, we shall learn that the thermal expansion coefficient of most glasses increases greatly in the glass transition region and beyond. Thus, the effects of temperature variation on the density of a glass are usually small in the glassy state. The densities of some single-component glasses are illustrated in Table 7.1. Of the various crystalline forms of silica, the density of fused silica (2.20 g/cm3) is closest to that of β-cristobalite, which is 2.25 g/cm3. The significant drop in the density of fused silica relative to that of α-quartz, which is the normal starting raw material, is indicative of the large structural changes that occur in the fusion process. The twisted hexagonal closepacked structure of α-quartz apparently “unwinds” and opens up. Densities of silica glasses derived from other processes are generally slightly lower, primarily because of impurities and voids. The density of B2O3 glass changes nonlinearly as a function of impurities such as OH, primarily because of the changes in boron coordination and structural groups. Densities of mixtures of glass formers generally vary monotonically between those of the end-members closely following the additivity principle, see, for example, SiO2-B2O3 and SiO2-GeO2 in Fig. 7.1 (plotted from data listed in Ref. [1]). The addition of network modifying components generally increases the density. This is to be expected, since the NWM ions attempt to occupy the Table 7.1 Densities (g/cm3) of single-component glasses at room temperature
SiO2 B2O3 GeO2 P2 O 5
2.20 1.800 3.628–3.65 >2.23
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Fig. 7.1 Density of glasses formed by mixtures of glass formers.
Fig. 7.2 Density of binary silicate glasses.
interstitial sites within the network. The variation of density with composition in several binary silicate glasses is shown in Fig. 7.2 (data listed in Ref. [1]). Of these, the data for Na2O-SiO2 and K2O-SiO2 glasses are shown in Fig. 7.3. Apparently, the density increases somewhat rapidly with initial additions of the alkali, and not as rapidly with continuing additions [2]. Note that despite the difference in ionic sizes between Na+ and K+, the density
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177
Fig. 7.3 Density of Na2O/K2O-SiO2 glasses.
apparently does not change significantly for similar molar content. Clearly, the larger weight of K+ is compensated by the larger partial molar volume of K2O as shown in Fig. 7.4. Fig. 7.4 also shows a sharp minimum at around 10–12 mol% added alkali. According to Bockris et al. [3], the initial additions of the alkali ions are completely “enclosed” within the SiOSi interstitial “cages.” At 10%–12%, the alkali ions begin to “open up” these cages. In the Li2O-SiO2 glasses (Fig. 7.5) with mol [Li2O] > 0.5 (i.e., [Li2O]/ [SiO2] > 1) range of “invert” compositions (Section 5.4), the density appears to decrease [4] and is related to the formation of Q2 (¼2 bridging oxygens per tetrahedron), Q1, and Q0. The density of alkali borates as a function of R (¼mol [R2O]/[B2O3]) is shown at room temperature [5] and at 900°C [6] in Fig. 7.6. Feller et al. [7] have shown that the packing fraction (¼volume of ions/volume of the polytope) in trigonal boron units remains relatively constant; it is substantially higher in the tetrahedrally coordinated boron units. Thus, density in borates tracks the fraction N4 of tetrahedrally coordinated borons, showing density maxima at 35–45 mol% added alkali (R ffi 0.5–0.8; Fig. 7.7). Again, by extracting the partial molar volumes of each of the components, it is
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Fig. 7.4 Partial molar volumes of components in alkali silicate glasses at 25°C. (After R.J. Callow, J. Soc. Glass Technol. 36 (1952) 137T. Reproduced with permission of the Society of Glass Technology.)
suggested [8] that the average B-B distance undergoes a minimum at varying alkali level (Fig. 7.8). The densities of several glasses with three or more components, including commercial systems, have been measured. Some of these are tabulated by Morey [9], Bansal and Doremus [1], and Mazurin et al. [10]. Molar volumes of glasses in the chalcogenide Ge-Sb-As-Se-Te system have been measured by Sreeram et al. [11] The values range between 17.6 and 18.5 cm3/mol. When plotted against the average coordination number, it is observed that there is a distinct minimum at < m> ¼ 2.40, suggesting an optimized packing at this value, which is in agreement with Phillips’s topological constraints model (Section 3.1.2). The molar volume increases with > 2.40 and appears to maximize at 2.67.
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179
Fig. 7.5 Density of Li2O-SiO2 glass. (Modified from A.M. Peters, F.M. Alamgir, S.W. Messer, S.A. Feller, K.L. Loh, Phys. Chem. Glasses 35 (1994) 212.)
Fig. 7.6 (A) Density of various alkali borate glasses at 25°C. (B) Density of various alkali borate melts at 900°C. (After H.C. Lim, S.A. Feller, J. Non-Cryst. Solids 94 (1987) 36. Reproduced with permission of Elsevier Science Publishers.)
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Fig. 7.7 Density and N4 in lithium borate glass. The solid line is the fraction N4 and solid points are the measured density, both plotted versus R. (Modified from S.A. Feller, N. Lower, M. Affatigato, Phys. Chem. Glasses 42 (2001) 240.)
Fig. 7.8 Average B-B distance in alkali borate melts at 1000°C. (After L. Shartsis, W. Capps, S. Spinner, J. Am. Ceram. Soc. 36 (1953) 35. Reproduced with permission of the American Ceramic Society.)
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181
Densities of a large number of heavy metal fluoride glasses have been measured. Some of these are tabulated by Drexhage [12]. The densities are generally in the 4.5–6.0 g/cm3 range.
7.4 Densification of glasses by high pressure or irradiation Bridgman and Simon [13] have shown that pressures higher than a threshold 100 kbars (¼10 GPa) caused fused silica glass to undergo compaction or irreversible densification. Densities increased about 7.5%, although a specimen with 17.5% density increase closely approaching the density of α-quartz was also found. Susman et al. [14] have obtained 20% increase in density of fused silica at 16 GPa pressure (slightly higher than that of α-quartz). It should be recognized that the density measurements above were done after pressure release and at ambient temperature. Apparently, the recovery of the density after pressure release is very sluggish for fused silica. On the other hand, for B2O3, no pressure threshold has been observed: pressure release causes a rapid steady recovery. Estimates of residual densification on pressure release at ambient temperature in various glasses vary from 6% to as much as 17% [15–17]. More recent measurements on densification in B2O3 at higher pressures using in situ measurement techniques [16, 17] have shown that the density measured at pressure increases steadily from 1.800 to as high as 3.15 g/cm3 at 14 GPa [18]. Density recovery is faster for all glasses when they are heated at temperatures close to their respective glass transition. There has been a considerable discussion over the exact mechanism of the densification and recovery, particularly over the question of the role of shear stresses (see Refs. [14, 16, 17, 19–21]). It is believed that the application of pressure causes a cooperative rotation of ionic polyhedra to move the atoms in a relatively open structure, such as that of a glass, closer to occupy some of the available interstitial volume. The rotation of the polyhedra causes the bonds such as SiOSi to bend without changing the near-neighbor separation. The presence of shear accelerates the pressure densification process. In B2O3 glass, the pressure-induced densification is accompanied by a conversion of B3 to B4. Accompanying these pressureinduced transformations, the fraction of borons associated with boroxol rings decreases from 75% to 40% at 2 GPa, and eventually to zero at 9 GPa (see Fig. 7.9). This is ascribed to the premise that in order for B3 to convert to B4, some of the boroxol rings in B2O3 (see Section 5.3) must disappear. A similar behavior is observed for soda lime borate glasses [22].
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Fig. 7.9 The percentage of boron atoms that are members of boroxol rings in B2O3 glass under pressure, as calculated from molecular dynamics simulations. (Reproduced from A. Zeidler et al., Phys. Rev. B 90 (2014) 024206.)
Irradiation of glasses by high-energy particles, such as fast neutrons, is also known to cause densification. Vitreous silica when irradiated with a neutron dosage of 2 1019 n/cm2 densifies 2%–3%. Further irradiation to 3 1020 n/cm2 causes a gradual dilatation whereby about 10% of the densification is recovered [23]. The densified structure shows a small increase in Young’s modulus and a small decrease in rigidity. The width of bond distance distribution (as measured by X-rays) increases slightly, implying the possibility of strains in the network. Gradual heating effects complete the recovery of the normal structure.
7.5 Calculation of density Because of the nature of glass as a “solution” of the various component oxides, ideas concerning the use of additivity factors for the calculation of the density ρ of glasses have been studied in great detail. The most successful of these are those by Huggins and Sun [24], which are based on the notion that the specific volumes of a component oxide are additive. Specific volume is the reciprocal of density, i.e., the volume of one gram of a substance. The specific volume of a component oxide is its contribution to the total specific volume. The accuracy within specific ranges of composition is often as good as 0.001 g/cm3. If the structural units within the glass can be
Density and molar volume
183
Table 7.2 Density difference factors Component Density change (g/cm3 per oxide 1 wt% oxide change)
SiO2 Al2O3 CaO MgO BaO Na2O K2O B2O3 ZnO PbO
0.0024 +0.0018 +0.0106 +0.0050 +0.0173 +0.0050 +0.0028 +0.0036 +0.0145 +0.0192
identified (as indicated in Section 7.3), then the calculation of density can be even more accurate using simple expressions of the type shown in Eq. (6.2). For soda lime silicate glasses, Elliott [25] has converted Huggins and Sun’s scheme to calculate the density changes on 1% addition of a given oxide. These density difference factors are illustrated in Table 7.2. The maximum error claimed is 0.0005 g/cm3. It may be interesting to note that, in a typical soda lime silicate glass, the substitution of 1 wt% Na2O by 1 wt% K2O leads to lowering of density. In closing this chapter, the student should note that because density of a glass is quite sensitive to its composition and cooling rate, and can be measured as well as calculated quite accurately, it is one of the most important quality control tools in a large-scale glass manufacturing operation.
7.6 Glass greats: W.E.S. Turner William Ernest Stephen Turner (1881–1963, Fig. 7.9) was a British chemist and a pioneer in glass technology. He joined the faculty at the University College of Sheffield in 1904. Recognizing the critical importance of glass technology at the beginning of World War I, he established the Department of Glass Manufacture at Sheffield in 1915. The following year it became the Department of Glass Technology, and Turner served as head of the department until his retirement in 1945. Turner’s impact on glass technology extended well beyond Sheffield. In 1916, he also founded the Society of Glass Technology to address the needs of industrial and academic glass research in the UK and was instrumental in the setting up of the International Commission on Glass in 1933, becoming its first elected president.
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Fig. 7.10 W.E.S. Turner (1881–1963), founder of the Society of Glass Technology.
Both the Society of Glass Technology and the International Commission on Glass have grown to be two of the most important and impactful organizations in the global glass community. Turner’s legacy lives on and has only grown over the decades since his passing (Fig. 7.10).
Summary Density ρ is defined as mass per unit volume. Appropriate units are: g/cm3 and kg m3. Molar volume ¼ formula wt/density. Partial molar volume vi ¼ (∂ V/∂ ni)nj, T, P. Hence, total molar volume V is the number-weighted sum of the partial molar volumes (“additivity”). Density is commonly measured using the Archimedes method by weighing a specimen in air and dividing it by the weight loss (buoyancy) upon immersion in water or some suitable fluid. One can also construct density gradient column with a heavy liquid such as methylene iodide (3.32 g/cm3) and a light liquid such as isopropyl salicylate (1.1 g/cm3). Density of a glass depends on the rate employed during cooling through the glass transition range. Faster cooled glasses are generally lighter than the slower cooled glasses. Density is very sensitive to glass composition and, hence, is one of the commonly used quality control methods in commercial glass production. Density of silica glass is 2.20 g/cm3. Density increases with the addition of NWM ions (lightest) Li < Na < K < Rb < Cs < Ag < Tl (heaviest). Fairly accurate density factors are available for many glass systems. Density of B2O3 glass is 1.800 g/cm3.
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185
Application of pressures >10 GPa begins to irreversibly compact silica glass. At about 16 GPa, density increase is as much as 20% (ρ > α-quartz!). B2O3 glass shows no threshold pressure; density increase by as much as 80% at 14 GPa pressure recovers rapidly after pressure release. It is believed that the reversibility is mostly a matter of kinetics (function of temperature). Irradiation with 2 1019 n/cm2 neutron dosage also densifies fused silica by about 2%–3%. Further irradiation causes 10% recovery.
Online resources (1) Room temperature glass density calculator: http://glassproperties. com/density/room-temperature/. (2) Video demonstrating the measurement of glass density: https://www. youtube.com/watch?v¼E2qvu5ugNXY. (3) TED-Ed talk on Archimedes’ principle: https://www.youtube.com/ watch?v¼ijj58xD5fDI.
Exercises (1) Show that when Elliott’s factors are converted to a mole basis, the density increase caused by 1 mol% addition of Na2O is still about 20% larger than that caused by 1 mol% addition of K2O despite the fact K+ is nearly double the size of Na+. What may qualitatively be concluded regarding the relative partial molar volumes of Na2O and K2O in silicate glasses? (2) Calculate the density of 15Na2O10CaO75SiO2 (mol%) glass using Huggins and Sun’s factors. Then, calculate its molar volume. (Ans: ρ ¼ 2.479 g/cm3. V ¼ 24.16 cm3.) (3) Calculate the change in density expected using Elliott’s factors by changing the glass composition of Exercise (2) to a 5Na2O10K2O10CaO75SiO2 (mol%) glass. (Ans: Δρ ¼ 0.006 g/cm3). (4) Imagine a glass plate of the composition 15Na2O10CaO75SiO2 (mol%) glass where 10 mol% of the surface Na2O has been replaced by K2O on an ion-for-ion basis (without electrostatic charge building up) well below the glass transition range. Would you expect any stresses to develop in the glass plate? What if Elliott’s density change factors for Na2O and K2O were exactly equal on a mole basis? (Note: Exercises 2, 3, and 4 form the basis of stress development by ion exchange in glass detailed in Section 14.8.1.)
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References [1] N.P. Bansal, R.H. Doremus, Handbook of Glass Properties, Chapter 4, Academic Press, 1986. [2] H. Doweidar, S.A. Feller, M. Affatigato, B. Tischendorf, C. Ma, E. Hammarsten, Phys. Chem. Glasses 40 (1999) 339. [3] J. O’M. Bockris, J.W. Tomlinson, J.L. White, Trans. Farad. Soc. 52 (1956) 299. [4] A.M. Peters, F.M. Alamgir, S.W. Messer, S.A. Feller, K.L. Loh, Phys. Chem. Glasses 35 (1994) 212. [5] H.C. Lim, S.A. Feller, J. Non-Cryst. Solids 94 (1987) 36. [6] E.F. Riebling, J. Am. Ceram. Soc. 50 (1967) 46. [7] S.A. Feller, N. Lower, M. Affatigato, Phys. Chem. Glasses 42 (2001) 240. [8] L. Shartsis, W. Capps, S. Spinner, J. Am. Ceram. Soc. 36 (1953) 35. [9] G.W. Morey, Properties of Glass, second ed., Reinhold, New York, 1954. [10] O.V. Mazurin, M.V. Streltsina, T.P. Shvaiko-Shvaikovskaya (Eds.), Handbook of Glass Data, Elsevier, Amsterdam, 1983. [11] A.N. Sreeram, A.K. Varshneya, D.R. Swiler, J. Non-Cryst. Solids 128 (1991) 294. [12] M.G. Drexhage, in: M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Tech., vol. 26, Academic Press, New York, 1985, pp. 151–243. [13] P.W. Bridgman, I. Simon, J. Appl. Phys. 24 (1953) 405. [14] S. Susman, K.J. Volin, R.C. Liebermann, G.D. Gwanmesia, Y. Wang, Phys. Chem. Glasses 31 (1990) 145. [15] D.R. Uhlmann, J.F. Hays, D. Turnbull, Phys. Chem. Glasses 8 (1967) 1. [16] H.M. Cohen, R. Roy, Phys. Chem. Glasses 6 (1965) 149. [17] W. Poch, Phys. Chem. Glasses 8 (1967) 129. [18] A. Zeidler, et al., Phys. Rev. B 90 (2014) 024206. [19] J.D. Mackenzie, J. Am. Ceram. Soc. 46 (1963) 461. [20] N. Mizouchi, A.R. Cooper, J. Am. Ceram. Soc. 56 (1973) 320. [21] J. Arndt, D. St€ offler, Phys. Chem. Glasses 10 (1969) 117. [22] M.M. Smedskjaer, R.E. Youngman, S. Striepe, M. Potuzak, U. Bauer, J. Deubener, H. Behrens, J.C. Mauro, Y. Yue, Sci. Rep. 4 (2014) 3770. [23] W. Primak, J. Phys. Chem. Sol. 13 (1960) 279. [24] M.L. Huggins, K.-H. Sun, J. Am. Ceram. Soc. 26 (1943) 4. [25] R.M. Elliott, J. Am. Ceram. Soc. 28 (1945) 303.
Density and molar volume 7.1 Definitions The density of a substance is defined as the mass per unit volume and is an intensive property. Appropriate units are g/cm3 or g/cc (in cgs system) and kg m3 in the SI system. The dependence of density on temperature is given by the volume thermal expansion coefficient. Since glasses may, in general, be regarded as solutions, a more useful property is the molar volume V, defined as the volume of 1 mol of glass. We then proceed to define partial molar volumes of the various structural units constituting the glass. The partial molar volume vi of a species i in a solution is defined by ∂V (7.1) vi ¼ ∂ni nj , T , P and hence, V¼
X
ni vi :
i
In essence, the total molar volume is treated as an extensive property in terms of the partial molar volumes of individual species or structural groups. The partial molar volume information can be extracted from the density data of glasses where the constituents have been systematically varied. If an Na+ ion is replaced by K+ in the glass and occupies the same site (without changing its size), it is clear that, although the glass density would increase because of the higher atomic weight of K+, the partial molar volume of the alkali ion would not change. Increase in the molar volume, if any, would be an increase in the partial molar volume of K+ relative to Na+. This is likely to have some relation to the relative ionic sizes of the two ions. One may readily note that, although the measurement of density changes in a family of glasses may provide only the trends, the extraction of the partial molar volumes from the density data provides further insight into structure while canceling out the effect of atomic masses.
Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00007-9
© 2019 Elsevier Inc. All rights reserved.
173
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7.2 Methods of measurement Density is measured most commonly by weighing a suitably selected piece of glass in air and dividing this weight by the buoyancy (the reduction in weight) when suspended in a fluid such as water. According to Archimedes’ principle, the buoyancy equals the weight of the displaced fluid. Thus, if wa is the weight of the specimen in air, and wb is the weight of the specimen when immersed in the fluid, then the buoyancy is wa wb and the density ρ ¼ ρmwa/(wa wb), where ρm is the density of the fluid (¼1.0 g/cm3 for water). As some glasses may interact with water, a suitable inert fluid such as kerosene may be selected. A kit supplied by scientific balance manufacturers can be used to obtain density measurements accurate to 0.001 g/cm3. Automatic measurements of density can be carried out using pycnometers which utilize He gas as the displacement fluid. A small sample, of the order of a few mg, is all that is needed to make the measurement. The accuracy of such instruments is generally about 0.002 g/cm3. In a manufacturing environment, a comparative density (easily accurate to 0.0002 g/cm3) is more rapidly obtained using a gradient column. The gradient column consists of a vertically held, one-end closed, long (often 1–1.5 m) glass tube containing a liquid with a linear density gradient. To prepare the density gradient, a heavy liquid such as sym-tetrabromoethane (ρ ¼ 2.96 g/cm3) or methylene iodide (ρ ¼ 3.32 g/cm3) is first poured into the tube and then a lighter liquid such as isopropyl salicylate (ρ ¼ 1.1 g/cm3) is gently poured to float above the heavy liquid.1 With time, the liquids diffuse into each other establishing a gradient with the help of the gravity. The gradient is calibrated by allowing density standards to gently float in the column at different heights. The density of an unknown glass is obtained by reading its floatation level against a pre-calibrated scale placed adjacent to the column. Another method also used commonly is the sink-float method, where the unknown specimen is gently dropped in a tube containing a slightly denser solution of organic liquids such as the ones stated above. The temperature around the tube is gradually changed until the previously floating specimen begins to sink. Calibration of the liquid’s density against temperature yields the density of the unknown. (See ASTM standard C729-75.)
1
Many of these liquids may pose health hazards. The reader is advised to check with the suppliers regarding their handling procedures.
Density and molar volume
175
7.3 Dependence on cooling rate, temperature, and composition As stated in Section 2.2, the volume variation upon cooling is fundamental to the transformation of a supercooled liquid into a glass. Except for a few anomalous cases, such as silica, a fast-cooled glass generally has a higher volume and, hence, lower density, relative to a slow-cooled glass of the same composition. Because density is inversely proportional to the volume, the fractional change in density with a ΔT change in temperature is given by 3αΔT, where α is the linear thermal expansion coefficient. Later, in Chapters 10 and 13, we shall learn that the thermal expansion coefficient of most glasses increases greatly in the glass transition region and beyond. Thus, the effects of temperature variation on the density of a glass are usually small in the glassy state. The densities of some single-component glasses are illustrated in Table 7.1. Of the various crystalline forms of silica, the density of fused silica (2.20 g/cm3) is closest to that of β-cristobalite, which is 2.25 g/cm3. The significant drop in the density of fused silica relative to that of α-quartz, which is the normal starting raw material, is indicative of the large structural changes that occur in the fusion process. The twisted hexagonal closepacked structure of α-quartz apparently “unwinds” and opens up. Densities of silica glasses derived from other processes are generally slightly lower, primarily because of impurities and voids. The density of B2O3 glass changes nonlinearly as a function of impurities such as OH, primarily because of the changes in boron coordination and structural groups. Densities of mixtures of glass formers generally vary monotonically between those of the end-members closely following the additivity principle, see, for example, SiO2-B2O3 and SiO2-GeO2 in Fig. 7.1 (plotted from data listed in Ref. [1]). The addition of network modifying components generally increases the density. This is to be expected, since the NWM ions attempt to occupy the Table 7.1 Densities (g/cm3) of single-component glasses at room temperature
SiO2 B2O3 GeO2 P2 O 5
2.20 1.800 3.628–3.65 >2.23
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Fig. 7.1 Density of glasses formed by mixtures of glass formers.
Fig. 7.2 Density of binary silicate glasses.
interstitial sites within the network. The variation of density with composition in several binary silicate glasses is shown in Fig. 7.2 (data listed in Ref. [1]). Of these, the data for Na2O-SiO2 and K2O-SiO2 glasses are shown in Fig. 7.3. Apparently, the density increases somewhat rapidly with initial additions of the alkali, and not as rapidly with continuing additions [2]. Note that despite the difference in ionic sizes between Na+ and K+, the density
Density and molar volume
177
Fig. 7.3 Density of Na2O/K2O-SiO2 glasses.
apparently does not change significantly for similar molar content. Clearly, the larger weight of K+ is compensated by the larger partial molar volume of K2O as shown in Fig. 7.4. Fig. 7.4 also shows a sharp minimum at around 10–12 mol% added alkali. According to Bockris et al. [3], the initial additions of the alkali ions are completely “enclosed” within the SiOSi interstitial “cages.” At 10%–12%, the alkali ions begin to “open up” these cages. In the Li2O-SiO2 glasses (Fig. 7.5) with mol [Li2O] > 0.5 (i.e., [Li2O]/ [SiO2] > 1) range of “invert” compositions (Section 5.4), the density appears to decrease [4] and is related to the formation of Q2 (¼2 bridging oxygens per tetrahedron), Q1, and Q0. The density of alkali borates as a function of R (¼mol [R2O]/[B2O3]) is shown at room temperature [5] and at 900°C [6] in Fig. 7.6. Feller et al. [7] have shown that the packing fraction (¼volume of ions/volume of the polytope) in trigonal boron units remains relatively constant; it is substantially higher in the tetrahedrally coordinated boron units. Thus, density in borates tracks the fraction N4 of tetrahedrally coordinated borons, showing density maxima at 35–45 mol% added alkali (R ffi 0.5–0.8; Fig. 7.7). Again, by extracting the partial molar volumes of each of the components, it is
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Fig. 7.4 Partial molar volumes of components in alkali silicate glasses at 25°C. (After R.J. Callow, J. Soc. Glass Technol. 36 (1952) 137T. Reproduced with permission of the Society of Glass Technology.)
suggested [8] that the average B-B distance undergoes a minimum at varying alkali level (Fig. 7.8). The densities of several glasses with three or more components, including commercial systems, have been measured. Some of these are tabulated by Morey [9], Bansal and Doremus [1], and Mazurin et al. [10]. Molar volumes of glasses in the chalcogenide Ge-Sb-As-Se-Te system have been measured by Sreeram et al. [11] The values range between 17.6 and 18.5 cm3/mol. When plotted against the average coordination number
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Fig. 7.5 Density of Li2O-SiO2 glass. (Modified from A.M. Peters, F.M. Alamgir, S.W. Messer, S.A. Feller, K.L. Loh, Phys. Chem. Glasses 35 (1994) 212.)
Fig. 7.6 (A) Density of various alkali borate glasses at 25°C. (B) Density of various alkali borate melts at 900°C. (After H.C. Lim, S.A. Feller, J. Non-Cryst. Solids 94 (1987) 36. Reproduced with permission of Elsevier Science Publishers.)
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Fig. 7.7 Density and N4 in lithium borate glass. The solid line is the fraction N4 and solid points are the measured density, both plotted versus R. (Modified from S.A. Feller, N. Lower, M. Affatigato, Phys. Chem. Glasses 42 (2001) 240.)
Fig. 7.8 Average B-B distance in alkali borate melts at 1000°C. (After L. Shartsis, W. Capps, S. Spinner, J. Am. Ceram. Soc. 36 (1953) 35. Reproduced with permission of the American Ceramic Society.)
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Densities of a large number of heavy metal fluoride glasses have been measured. Some of these are tabulated by Drexhage [12]. The densities are generally in the 4.5–6.0 g/cm3 range.
7.4 Densification of glasses by high pressure or irradiation Bridgman and Simon [13] have shown that pressures higher than a threshold 100 kbars (¼10 GPa) caused fused silica glass to undergo compaction or irreversible densification. Densities increased about 7.5%, although a specimen with 17.5% density increase closely approaching the density of α-quartz was also found. Susman et al. [14] have obtained 20% increase in density of fused silica at 16 GPa pressure (slightly higher than that of α-quartz). It should be recognized that the density measurements above were done after pressure release and at ambient temperature. Apparently, the recovery of the density after pressure release is very sluggish for fused silica. On the other hand, for B2O3, no pressure threshold has been observed: pressure release causes a rapid steady recovery. Estimates of residual densification on pressure release at ambient temperature in various glasses vary from 6% to as much as 17% [15–17]. More recent measurements on densification in B2O3 at higher pressures using in situ measurement techniques [16, 17] have shown that the density measured at pressure increases steadily from 1.800 to as high as 3.15 g/cm3 at 14 GPa [18]. Density recovery is faster for all glasses when they are heated at temperatures close to their respective glass transition. There has been a considerable discussion over the exact mechanism of the densification and recovery, particularly over the question of the role of shear stresses (see Refs. [14, 16, 17, 19–21]). It is believed that the application of pressure causes a cooperative rotation of ionic polyhedra to move the atoms in a relatively open structure, such as that of a glass, closer to occupy some of the available interstitial volume. The rotation of the polyhedra causes the bonds such as SiOSi to bend without changing the near-neighbor separation. The presence of shear accelerates the pressure densification process. In B2O3 glass, the pressure-induced densification is accompanied by a conversion of B3 to B4. Accompanying these pressureinduced transformations, the fraction of borons associated with boroxol rings decreases from 75% to 40% at 2 GPa, and eventually to zero at 9 GPa (see Fig. 7.9). This is ascribed to the premise that in order for B3 to convert to B4, some of the boroxol rings in B2O3 (see Section 5.3) must disappear. A similar behavior is observed for soda lime borate glasses [22].
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Fig. 7.9 The percentage of boron atoms that are members of boroxol rings in B2O3 glass under pressure, as calculated from molecular dynamics simulations. (Reproduced from A. Zeidler et al., Phys. Rev. B 90 (2014) 024206.)
Irradiation of glasses by high-energy particles, such as fast neutrons, is also known to cause densification. Vitreous silica when irradiated with a neutron dosage of 2 1019 n/cm2 densifies 2%–3%. Further irradiation to 3 1020 n/cm2 causes a gradual dilatation whereby about 10% of the densification is recovered [23]. The densified structure shows a small increase in Young’s modulus and a small decrease in rigidity. The width of bond distance distribution (as measured by X-rays) increases slightly, implying the possibility of strains in the network. Gradual heating effects complete the recovery of the normal structure.
7.5 Calculation of density Because of the nature of glass as a “solution” of the various component oxides, ideas concerning the use of additivity factors for the calculation of the density ρ of glasses have been studied in great detail. The most successful of these are those by Huggins and Sun [24], which are based on the notion that the specific volumes of a component oxide are additive. Specific volume is the reciprocal of density, i.e., the volume of one gram of a substance. The specific volume of a component oxide is its contribution to the total specific volume. The accuracy within specific ranges of composition is often as good as 0.001 g/cm3. If the structural units within the glass can be
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Table 7.2 Density difference factors Component Density change (g/cm3 per oxide 1 wt% oxide change)
SiO2 Al2O3 CaO MgO BaO Na2O K2O B2O3 ZnO PbO
0.0024 +0.0018 +0.0106 +0.0050 +0.0173 +0.0050 +0.0028 +0.0036 +0.0145 +0.0192
identified (as indicated in Section 7.3), then the calculation of density can be even more accurate using simple expressions of the type shown in Eq. (6.2). For soda lime silicate glasses, Elliott [25] has converted Huggins and Sun’s scheme to calculate the density changes on 1% addition of a given oxide. These density difference factors are illustrated in Table 7.2. The maximum error claimed is 0.0005 g/cm3. It may be interesting to note that, in a typical soda lime silicate glass, the substitution of 1 wt% Na2O by 1 wt% K2O leads to lowering of density. In closing this chapter, the student should note that because density of a glass is quite sensitive to its composition and cooling rate, and can be measured as well as calculated quite accurately, it is one of the most important quality control tools in a large-scale glass manufacturing operation.
7.6 Glass greats: W.E.S. Turner William Ernest Stephen Turner (1881–1963, Fig. 7.9) was a British chemist and a pioneer in glass technology. He joined the faculty at the University College of Sheffield in 1904. Recognizing the critical importance of glass technology at the beginning of World War I, he established the Department of Glass Manufacture at Sheffield in 1915. The following year it became the Department of Glass Technology, and Turner served as head of the department until his retirement in 1945. Turner’s impact on glass technology extended well beyond Sheffield. In 1916, he also founded the Society of Glass Technology to address the needs of industrial and academic glass research in the UK and was instrumental in the setting up of the International Commission on Glass in 1933, becoming its first elected president.
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Fig. 7.10 W.E.S. Turner (1881–1963), founder of the Society of Glass Technology.
Both the Society of Glass Technology and the International Commission on Glass have grown to be two of the most important and impactful organizations in the global glass community. Turner’s legacy lives on and has only grown over the decades since his passing (Fig. 7.10).
Summary Density ρ is defined as mass per unit volume. Appropriate units are: g/cm3 and kg m3. Molar volume ¼ formula wt/density. Partial molar volume vi ¼ (∂ V/∂ ni)nj, T, P. Hence, total molar volume V is the number-weighted sum of the partial molar volumes (“additivity”). Density is commonly measured using the Archimedes method by weighing a specimen in air and dividing it by the weight loss (buoyancy) upon immersion in water or some suitable fluid. One can also construct density gradient column with a heavy liquid such as methylene iodide (3.32 g/cm3) and a light liquid such as isopropyl salicylate (1.1 g/cm3). Density of a glass depends on the rate employed during cooling through the glass transition range. Faster cooled glasses are generally lighter than the slower cooled glasses. Density is very sensitive to glass composition and, hence, is one of the commonly used quality control methods in commercial glass production. Density of silica glass is 2.20 g/cm3. Density increases with the addition of NWM ions (lightest) Li < Na < K < Rb < Cs < Ag < Tl (heaviest). Fairly accurate density factors are available for many glass systems. Density of B2O3 glass is 1.800 g/cm3.
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Application of pressures >10 GPa begins to irreversibly compact silica glass. At about 16 GPa, density increase is as much as 20% (ρ > α-quartz!). B2O3 glass shows no threshold pressure; density increase by as much as 80% at 14 GPa pressure recovers rapidly after pressure release. It is believed that the reversibility is mostly a matter of kinetics (function of temperature). Irradiation with 2 1019 n/cm2 neutron dosage also densifies fused silica by about 2%–3%. Further irradiation causes 10% recovery.
Online resources (1) Room temperature glass density calculator: http://glassproperties. com/density/room-temperature/. (2) Video demonstrating the measurement of glass density: https://www. youtube.com/watch?v¼E2qvu5ugNXY. (3) TED-Ed talk on Archimedes’ principle: https://www.youtube.com/ watch?v¼ijj58xD5fDI.
Exercises (1) Show that when Elliott’s factors are converted to a mole basis, the density increase caused by 1 mol% addition of Na2O is still about 20% larger than that caused by 1 mol% addition of K2O despite the fact K+ is nearly double the size of Na+. What may qualitatively be concluded regarding the relative partial molar volumes of Na2O and K2O in silicate glasses? (2) Calculate the density of 15Na2O10CaO75SiO2 (mol%) glass using Huggins and Sun’s factors. Then, calculate its molar volume. (Ans: ρ ¼ 2.479 g/cm3. V ¼ 24.16 cm3.) (3) Calculate the change in density expected using Elliott’s factors by changing the glass composition of Exercise (2) to a 5Na2O10K2O10CaO75SiO2 (mol%) glass. (Ans: Δρ ¼ 0.006 g/cm3). (4) Imagine a glass plate of the composition 15Na2O10CaO75SiO2 (mol%) glass where 10 mol% of the surface Na2O has been replaced by K2O on an ion-for-ion basis (without electrostatic charge building up) well below the glass transition range. Would you expect any stresses to develop in the glass plate? What if Elliott’s density change factors for Na2O and K2O were exactly equal on a mole basis? (Note: Exercises 2, 3, and 4 form the basis of stress development by ion exchange in glass detailed in Section 14.8.1.)
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References [1] N.P. Bansal, R.H. Doremus, Handbook of Glass Properties, Chapter 4, Academic Press, 1986. [2] H. Doweidar, S.A. Feller, M. Affatigato, B. Tischendorf, C. Ma, E. Hammarsten, Phys. Chem. Glasses 40 (1999) 339. [3] J. O’M. Bockris, J.W. Tomlinson, J.L. White, Trans. Farad. Soc. 52 (1956) 299. [4] A.M. Peters, F.M. Alamgir, S.W. Messer, S.A. Feller, K.L. Loh, Phys. Chem. Glasses 35 (1994) 212. [5] H.C. Lim, S.A. Feller, J. Non-Cryst. Solids 94 (1987) 36. [6] E.F. Riebling, J. Am. Ceram. Soc. 50 (1967) 46. [7] S.A. Feller, N. Lower, M. Affatigato, Phys. Chem. Glasses 42 (2001) 240. [8] L. Shartsis, W. Capps, S. Spinner, J. Am. Ceram. Soc. 36 (1953) 35. [9] G.W. Morey, Properties of Glass, second ed., Reinhold, New York, 1954. [10] O.V. Mazurin, M.V. Streltsina, T.P. Shvaiko-Shvaikovskaya (Eds.), Handbook of Glass Data, Elsevier, Amsterdam, 1983. [11] A.N. Sreeram, A.K. Varshneya, D.R. Swiler, J. Non-Cryst. Solids 128 (1991) 294. [12] M.G. Drexhage, in: M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Tech., vol. 26, Academic Press, New York, 1985, pp. 151–243. [13] P.W. Bridgman, I. Simon, J. Appl. Phys. 24 (1953) 405. [14] S. Susman, K.J. Volin, R.C. Liebermann, G.D. Gwanmesia, Y. Wang, Phys. Chem. Glasses 31 (1990) 145. [15] D.R. Uhlmann, J.F. Hays, D. Turnbull, Phys. Chem. Glasses 8 (1967) 1. [16] H.M. Cohen, R. Roy, Phys. Chem. Glasses 6 (1965) 149. [17] W. Poch, Phys. Chem. Glasses 8 (1967) 129. [18] A. Zeidler, et al., Phys. Rev. B 90 (2014) 024206. [19] J.D. Mackenzie, J. Am. Ceram. Soc. 46 (1963) 461. [20] N. Mizouchi, A.R. Cooper, J. Am. Ceram. Soc. 56 (1973) 320. [21] J. Arndt, D. St€ offler, Phys. Chem. Glasses 10 (1969) 117. [22] M.M. Smedskjaer, R.E. Youngman, S. Striepe, M. Potuzak, U. Bauer, J. Deubener, H. Behrens, J.C. Mauro, Y. Yue, Sci. Rep. 4 (2014) 3770. [23] W. Primak, J. Phys. Chem. Sol. 13 (1960) 279. [24] M.L. Huggins, K.-H. Sun, J. Am. Ceram. Soc. 26 (1943) 4. [25] R.M. Elliott, J. Am. Ceram. Soc. 28 (1945) 303.