The viscosity of glass

CHAPTER 9

The viscosity of glass 9.1 Introduction Viscosity is a liquid-state property. The application of a shear force causes the atoms and molecular groups to undergo displacement with respect to each other, which continues with time so long as the application of force continues. Viscosity is the inverse of fluidity and is a measure of the resistance to shear deformation with time. The time rate of deformation e_ xy is linearly related to the shearing stress σ xy through Newton’s law of viscosity: σ xy ¼ η_e xy

(9.1)

where η is the coefficient of viscosity. (The reader should note the use of “” to denote the derivative with respect to time.) The coefficient of viscosity, or simply the viscosity, has the dimensions of pressure multiplied by time. The SI unit for viscosity, η, is Pa s (since strain is unitless). When the stress is written in units of dyn/cm2 instead of Pa, the unit of viscosity is called Poise. In the old literature and in many parts of the glass industry, the most commonly used unit is Poise (often abbreviated as “P”). However, the Pa s unit is the accepted SI unit and should be used for all future studies. The conversion between Pa s and Poise is trivial: 1 Pa s ¼ 10 Poise. Note the similarity between Eq. (9.1) and Eq. (8.4) for shear modulus, which is σ xy ¼ Gexy. When a shear stress acts on an elastic body, the body undergoes an instantaneous equilibrium shape change, the shear strain is related to the shear stress through Eq. (8.4). On the other hand, when a shear stress acts on a viscous body, the body starts flowing or deforming with time at a rate given by Eq. (9.1). In essence, a viscous body is one which exhibits a plastic yield strength approaching zero. Continuing application of a shearing stress causes it to exceed its yield strength, hence, deform continuously. It is therefore reasonable to suspect that the solutions to many viscosity problems can be found simply by obtaining a solution to an analogous elasticity problem, replacing the shear strain by shear strain rate and writing η in place of G. This viscosity-elasticity analogy happens to be true for highly viscous bodies; however, it requires that the body is isotropic and incompressible. For an incompressible solid, the Poisson ratio, ν ¼ 0.5, and the Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00009-2

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Young’s modulus E ¼ 3G (cf. Eq. 8.7). A formal statement of this analogy has been made by Goodier [1] as follows: “Let there be two isotropic and homogeneous bodies of the same shape, one purely viscous, the other purely elastic and incompressible. The two are either loaded in the same way, or, if certain displacements are prescribed as given functions in the elastic body, then the corresponding velocities in the viscous bodies are prescribed as the same functions. Let the displacements of the incompressible elastic body be expressed as certain functions of x, y, z, and the modulus of rigidity G. Then the velocities of the viscous body are expressed by the same functions when the coefficient of viscosity η is substituted for G as long as the deformation makes no significant change in the boundaries. The converse will also be true.” The reader may note that the analogy is not confined to just the shear stresses. Elastic equations involving any type of stress can be converted to viscous equations by substituting η for G, 0.5 for ν, and writing strain rates in place of strains. Trouton [2] was perhaps the first to note this analogy for the simple case of the extension of pitch under uniaxial load. Trouton defined λ, the coefficient of viscous traction, through the equation: σ x ¼ λ_e x

(9.2)

and showed that λ  3η, which is also predicted by analogy to the elastic situation described by σ x ¼ Eex. Like the elasticity situation, Eq. (9.1) represents linear viscosity where η does not depend on the strain rate. Fluids that obey Eq. (9.1) are termed Newtonian fluids. Owing to this linearity, the Boltzmann superposition principle discussed in Chapter 13 and at the end of the Appendix is also applicable. On an atomic scale, the fluidity could be ascribed to the motion of the interstitial spaces or “holes” of the size of the molecules. A hole may be thought of as a portion of the empty space which passed into the liquid (concurrent with a molecule which detached itself from the interior of the liquid and came to the surface occupying the empty space). Since the molecule has a large entropy in the empty space, we may also argue that so does the hole in a liquid. The high entropy of the hole implies a large number of rearrangements within the liquid, and hence the entropy contribution of the holes is termed configurational entropy. In a liquid at a given temperature, the holes have certain distribution of energies which enables them to overcome potential barriers with a certain probability distribution. In other words, the holes are able to move around with random velocities with various orientations, which are representations of the molecular velocity at that temperature. The lack of large holes in a crystal gives rise to a well-defined melting point.

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The surface atoms must move significantly before releasing underlying atoms, sequentially, out of their well-defined potential. Glass and glass-forming melts, on the other hand, are presumed to have a large number of holes, collectively called the free volume, already in the structure. With increasing temperature, the holes can move and grow significantly (hence, the “shape” of the body changes) without bringing an appreciable change in the neighborhood potentials. An analogy here to a packed and a less-filled elevator is close. Crystals act like a packed elevator where people cannot move until the people at the door have moved out at the instant of opening the door. Glass and glass-forming melts act like a less-filled elevator where individual motion is possible at all times, and more so when the door opens. In the seemingly solid state, the skeleton of glass made up of the networkforming ions does not develop a significant shear-strain rate when ordinary magnitudes of shearing stresses are applied. Such ions do not have alternative equilibrium sites toward which to migrate. However, it is possible for the network modifying ions such as Na+, H+, and OH– to find alternative sites to move to upon application of shearing stresses. The resulting apparent viscosity in the seemingly solid state manifests itself in the form of internal friction—a subject treated in Section 13.2.1 in more detail. Because of the subsequent rearrangement of holes, some dimensional changes with time of shearing are likely. With the release of the shearing stresses, the network modifying ions generally return to their original sites causing glass to recover its original shape. One may refer to this behavior as delayed elasticity (also called anelasticity)—a topic treated further in Section 13.2.3 because of its prominence as the mass approaches the glass transition range. The reader is referred to Douglas [3] whose pioneering work on viscosity of glass has done much to clarify our understanding.

9.2 Viscosity reference points As pointed out in connection with the V–T diagram (Fig. 2.1), a glassforming melt acts as a liquid at high temperatures and effectively turns into a solid on cooling. This η–T relationship for a typical soda lime container glass is shown in Fig. 9.1. At around 1450°C, where the glass is “melted,” the viscosity is around 10Pa s. (For those who might have been in the moonshine business, this is roughly the viscosity of hot molasses. The melt is a thick syrup at this point. Also, for comparison, the room temperature viscosity of water is 0.001 Pa s and that of glycerine about 1 Pas.) The melt can flow, but not readily. As the temperature decreases, the viscosity increases monotonically by several orders of magnitude. At around 750°C, the viscosity

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Fig. 9.1 Variation of the viscosity of a common soda lime silicate glass with temperature.

is about 106.6 Pas, at which point the mass would slump under its own weight. Below about 450°C, the viscosity of glass is higher than about 1015 Pas, at which point it becomes difficult to distinguish the glass from a solid form. It should be noted that there are no discontinuities in this curve, which is the key to continuous glass forming discussed in Chapter 22. Because of the importance of viscosity to commercial processes, there are at least four standard viscosity reference points utilized in common practice. These are: The working point 103 Pa s ¼ 104 Poise The softening point 106.65 Pa s ¼ 107.65 Poise The annealing point 1012 Pa s ¼ 1013 Poise The strain point 1013.5 Pa s ¼ 1014.5 Poise In addition, melting point (viscosity ¼ 10 Pa s), and flow point (viscosity ¼ 104 Pa s) are also used occasionally.

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As one would expect, the working point is the viscosity at which a mass of glass is delivered to a machine to be worked on, and the softening point is the minimum viscosity that the worked mass of glass must have before being allowed to stand on its own (else it will deform under its own weight). The temperature interval between the working point and the softening point is called the working range. The usefulness of the viscosity reference points lies in the specification of temperatures at which the viscosity is a certain value. The softening point was defined by Littleton [4] as the temperature at which a glass fiber, 0.55–0.75 mm diameter and 23.5 cm long of which the top 10 cm is heated at a rate of 5°C/min when suspended vertically in a furnace of specified characteristics, elongates under its own weight at a rate of 1 mm/min. The annealing point and the strain point approximately bracket the upper and the lower end, respectively, of the glass transformation range used for practical annealing of glass products. (In glass technology, the term “annealing” refers to releasing stresses which develop during the glassforming process and is discussed in detail in Chapter 13.) A useful definition of the annealing point is “the temperature at which a glass would release 95% of its stresses within 15 min.” At the strain point, the stress release occurs over roughly 6 h. It is indicated later in Chapter 13 that these definitions are best stated as “stress release occurs within a matter of minutes at the annealing point” and “within a matter of hours at the strain point.”

9.3 Measurement of viscosity There is no single technique that may be utilized to measure the viscosity of glass over the entire range spanning roughly 14 orders of magnitude change. The measurements must be made using various techniques applicable for limited ranges. These are summarized in the following table: Range

Method

Viscosity range

Melting

Stokes’ falling sphere/bubble rise Margules rotating cylinder 8 Parallel plate > > > > < Penetration viscometer Fiber elongation > > Beam bending > > : Disappearance of stress

<104 Pa s

Softening and annealing

<106.5 Pa s 105 Pa s < η < 109 Pa s 106 Pa s < η < 1011 Pa s 105 Pa s < η < 1015.5 Pa s 107 Pa s < η < 1014 Pa s 1011 Pa s < η < 1014 Pa s

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Measurement of viscosities higher than about 1010 Pa s requires a waiting period for glass stabilization. (The origin of this problem is also discussed in Chapter 13.) The waiting time is generally of the order of 3η/G (recall, G is the shear modulus). Since G  30 GPa for many silicate glasses, viscosities larger than about 1015.5 Pa s cannot be accurately measured as a practicality. Falling sphere and bubble rise methods are based on Stokes’ formula. According to Stokes, a sphere of radius r moving with a velocity v through a viscous medium of viscosity η experiences a frictional force F given by F ¼ 6πrην

(9.3)

In the falling sphere method, a platinum ball of radius r is allowed to fall freely in the molten glass and its velocity v of fall is measured. Application of Stokes’ formula gives v ¼ 2r 2 ðρ  ρo Þg=9η

(9.4)

where g is the acceleration due to gravity. In the rise of a fluid bubble method, a bubble of a known gas is introduced in the molten glass, and the velocity of rise is monitored. A correction for the viscosity of the fluid inside the bubble is applied to the Stokes’ formula as   2r 2 g 3η + 3η0 (9.5) ðρ  ρ0 Þ v¼ 2η + 3η0 9η where ρ and η are the density and the viscosity, respectively, of the glass, ρ0 and η0 are the corresponding values for the fluid inside the bubble, and r is the radius of the bubble. There are three variations employed for the rotating cylinder method. In the original Margules [5] version, shown in Fig. 9.2, the molten glass is filled in the space between two concentric platinum cylinders. The outer cylindrical crucible is rotated at a constant angular velocity, and the twist angle of the inner spindle is measured. The method is suitable for measuring viscosities <104 Pa s. To extend the measurement range to about 106.5 Pa s, a second variation due to Lillie [6] includes an aperiodic oscillation. The inner platinum spindle is displaced from its equilibrium position using an electric field coil and allowed to return to its original position as an over-damped harmonic oscillation under the force of a torsion wire is used to suspend the inner spindle. The time δt to pass through twist angles θ1 and θ2 (<θ1) is related to the viscosity using the equation: η¼

Aδt lnðθ1 =θ2 Þ

(9.6)

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Fig. 9.2 Rotating cylinder viscometer. (After H. Rawson, Properties and Applications of Glass, Elsevier Science Publishers, 1980, p. 44. Adapted after A. Dietzel, R. Br€ uckner, Glastech. Ber. 28 (1955) 455. Reproduced with permission of Elsevier Science Publishers.)

where A is a constant for the apparatus obtained using a standard calibration glass. It should be noted that this method can be applied to measuring viscosities down to a fraction of a Pa s. The third variation is Searle’s viscometer [7], where a platinum inner spindle is rotated at a constant velocity in the molten glass contained in a stationary outer platinum crucible. Viscosity is calculated from the torque M applied to the cylinder to maintain a given velocity of rotation ω using the equation:   M Ro2  Ri2 η¼ , (9.7) 4πLωRo2 Ri2 where Ro and Ri are the radii of the outer and inner cylinders, respectively, and L is the immersion depth. Some authors have claimed to use rotational methods for

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viscosities as high as 109 Pa s. In the authors’ experience, very high viscosity fluids tend to wrap around the inner spindle, causing erroneous measurements. Rotational viscometry methods are described in detail by Napolitano et al. [8] and Rawson [9]. In all these techniques, one measures the viscosity of a standard glass (e.g., procured from the National Institute for Standards and Technology) to calibrate the instrument. Non-concentricity of the rotating cylinders, bubbles trapped in the fluid, insufficient immersion depth of the inner spindle, and temperature nonuniformity are the key sources of error. In a cylindrical rod penetration viscometer, a rod of radius r and mass m is allowed to penetrate the glass to a given depth h. Viscosity is calculated from the rate of penetration dh/dt using the equation [10]: 2P η¼ (9.8) 9πr ðdh=dt Þ which is basically the Stokes’ formula (9.3) having a constant of 4.5 in place of 6. The load (P) is the sum of the gravitational force (mg) and any applied force on the rod. In the ball penetration viscometer method [10, 11], a hemispherical ball of radius r is loaded (load ¼ P) and allowed to penetrate to a distance h into the glass. In this case, viscosity is given by 3P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9.9) η¼ 16ðdh=dt Þ hð2r  hÞ The fact that the contact surface of glass does not usually follow the indenter geometry (see Section 8.3.1) may cause the indenter methods to be quite inaccurate. The parallel plate viscometer is described in detail by Varshneya et al. [12]. The schematics of the setup are shown in Fig. 9.3. A disk of glass, roughly 6–8 mm diameter  3–5 mm thick, is sandwiched between two parallel plates inside a well-insulated furnace. Direct contact with the plate material is avoided by employing thin Pt foil or alumina substrates. The upper pedestal is loaded, and the rate of sagging dh/dt is recorded as a function of time. The primary source of error in this technique is the slip condition between the glass specimen and the substrates. For a perfect “no slip” condition (approximately valid when using alumina substrates), the viscosity is given by 2πPh5 (9.10) 3V ð2πh3 + V Þðdh=dtÞ where V is the volume of the cylindrical disk specimen of height h. The fiber softening method is one of the most important of all the viscosity measurement techniques. A 0.55–0.75 mm diameter glass fiber is η¼

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Fig. 9.3 Schematic diagram of a parallel plate viscometer.

drawn. One end is fused to make a ball. The fiber is cut to 23.5 cm length, and then suspended inside a specified furnace, which only covers the top 10 cm (Fig. 9.4). The furnace is heated at 5°C/min. The sagging of the lower end is sighted using a telescope and measured as a function of time. If p is the length of the fiber inside the furnace and L is the length outside, then η¼

ðLp + p2 =2Þρg  γp=r 3ðdp=dt Þ

(9.11)

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Fig. 9.4 Schematic diagram of the Littleton softening point method.

where r is the average radius, ρ and γ are the density and the surface tension of the glass, and g is the acceleration due to gravity. (It may be noted that the term γp/r arises due to surface energy and attempts to shrink the fiber. Hence, L and p may be chosen such that the fiber neither elongates nor shrinks. This would then allow a measurement of γ of the glass.) One plots the rate of elongation versus the temperature. The temperature at which dp/dt ¼ 1 mm/min is identified as the softening point. Because of the varying ρ and γ, the actual value of the viscosity varies somewhat from glass to glass. For a soda lime glass, dp/dt ¼ 1 mm/min corresponds to roughly 106.65 Pa s viscosity (see also ASTM standard C338-73 for the measurement of the softening point). To measure high viscosities, one may need to load the fiber by placing weights in a pan attached to the lower end. Viscosities in this manner have been measured to the practically highest limit of about 1017 Pa s.

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In the beam bending method, a simply supported beam is centrally loaded and its rate of sag at the center is recorded. The equation to calculate the viscosity is derived by extending viscoelastic analogy (Section 9.1) to the three-point elastic beam bending method (Eq. 8.8) and is left as an exercise for the reader to solve. From the following section, it should be clear to the student that the measurement of temperature close to the specimen is vital to all viscosity measurements.

9.4 Viscosity of some common glasses A plot of log η versus 1/T for various glass-forming oxides is shown in Fig. 9.5. The measured activation energies are shown in Table 9.1. The large difference between the viscosities of silica and boric oxide glasses is understandable since silica is a fully connected tetrahedral network, whereas boric oxide is only trigonally connected. The difference between the tetrahedrally coordinated germania and silica viscosities is not easily understood. Presumably there are subtle differences between the two, for instance, one of the four oxygens in germania is farther away from Ge relative to the other three. We might also imagine that, in single component oxide glasses, the activation barrier for viscous flow, ΔH,

Fig. 9.5 Variation of log(viscosity) versus 10,000/T for simple glass-forming oxides. (After R.H. Doremus, Glass Science, John Wiley & Sons, New York, 1973, p. 105. Reproduced with permission of the publishers.)

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Table 9.1 Activation energies for viscous flow in some simple glasses Glass Activation energy (kcal/mol) Temperature range (°C)

Vitreous silica Vitreous germania Vitreous P2O5 Vitreous B2O3

170 123 75 41.5 83–12

1100–1400 1600–2500 540–1500 545–655 26–1300

might involve the cation-oxygen bond breaking. Since the single bond strengths of the glass-forming cation and oxygen are >380 kJ/mol (90 kcal/mol), one would expect the activation energies to be of the same order. Yet, the data shown in Table 9.1 appear to vary from oxide to oxide; they are particularly low in trigonally connected B2O3. It is clear that fluidity may not involve a mere single bond breaking; there may be double bondbreaking events in addition to bond bending in other instances. The observed activation energy does vary with temperature. In B2O3 glass, the activation barrier ΔH in the transformation range is about 325 kJ/ mol and decreases to only about 57 kJ/mol at fairly high temperatures. The influence of water impurity on silica glass is shown in Fig. 9.6A–D. The rather dramatic effect is true of nearly all oxide glasses [13]. Fig. 9.6D, taken from Ref. [13b], shows that the glass transition temperature, Tg, of many water-containing silicate glasses varies along a master curve when plotGN ted as reduced glass transition temperature (¼Tg/TGN is the Tg g , where Tg of 0.02 wt% water containing glass) versus the total water content. Since viscosity is of the utmost importance to glass-forming processes, the practical significance of water impurity in glass is overwhelming. Commercial soda lime silicate glasses generally contain around 350 ppm. Water in glass is primarily derived from raw materials as surface adsorbed moisture and as chemically bound water of crystallization. It may also be derived from the combustion products of the fuel used to melt the glass. Additions of alkali greatly reduce the viscosity of the tetrahedrally connected networks. Such effect, however, is nonlinear with respect to the composition. The initial additions reduce the viscosity much more than the later ones (Fig. 9.7A). (The observation implies that additivity factors for viscosity apply over only a narrow range of compositions.) When the alkali additions to an alkali silicate glass are plotted on a mol/cm3 basis, the data for Li2O, Na2O, and K2O fall on the same curve (Fig. 9.7B). Mixing alkalis generally lowers the viscosity of alkali silicate glasses at a given temperature, if the viscosities of the end members are relatively close. Fig. 9.8 plots the isokom (equal viscosity) temperatures for xNa2O(20  x)

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Fig. 9.6 Effect of water on the viscosity of glasses. (A) Variation of Tg of SiO2. (B) Temperature dependence of viscosity for Na2OCaOSiO2. OH content: A ¼ 110 ppm, B ¼ 356 ppm, and C ¼ 763 ppm. (C) Comparison of changes in Tg of Na2OK2OZnOAl2O3SiO2 glass with the addition of H2O or Na2O. (D) Reduced glass transition temperature versus total water content for several hydrous silicate glasses. ((A) After J.E. Shelby, G.L. McVey, J. Non-Cryst. Solids 20 (1976) 439. (B) After P.W. McMillan, A. Chlebnik, J. Non-Cryst. Solids 38–39 (1980) 509. (C) Modified from R.F. Bartholomew, in: M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Technol., vol. 22, Academic Press, 1982, pp. 75–127. (D) Modified from J. Deubener, R. M€ uller, H. Behrens, G. Heide, J. Non-Cryst. Solids 330 (2003) 268.) Continued

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Fig. 9.6, cont‘d

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Fig. 9.7 (A) Viscosity of melts in binary silicate and germanate systems. (B) Viscosity of alkali silicate glasses as a function of the added mol/cc alkali. ▲ K2O;  and 4 Na2O; ● Li2O. (After H. Rawson, Properties and Applications of Glass, Elsevier Science Publishers, Amsterdam, Holland, 1980, p. 64. Reproduced with permission of the publisher.)

K2O80SiO2 glasses (data from Nemilov [15]). This behavior, which is attributed to the phenomenon called the mixed alkali effect (Chapter 14), is less pronounced with lower total alkali, or mixed Li2O/K2O-containing glasses. Nearly all other third component additions to alkali silicates increase the viscosity as shown, for instance, in Fig. 9.9 for the gradual replacement of Na2O by CaO, MgO, or Al2O3 in a 2Na2O6SiO2 glass. In alkali borate melts, the viscosity versus alkali content plots show pronounced maxima at low temperatures (Fig. 9.10A taken from Visser and Stevels [14]). The magnitude of these maxima tends to reduce with increasing temperature. This observation is one of the two primary features of the boron anomaly (Section 5.6). Like the silicates, mixed-alkali borate glasses also show negative deviation relative to that displayed by the single alkali borates. However, the mixed alkali effect does not interfere with the concurrent

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Fig. 9.8 Isokom temperatures for xNa2O(20  x)K2O80SiO2 glasses. Log η(Poise) ¼ 8.0 (); 10.0 (); 12.0 (4); 16.0 (R). (Data are from S.V. Nemilov, J. Appl. Chem. USSR Eng. Trans. 42 (1969) 46.)

Fig. 9.9 Effect on the viscosity (Poise) at 1000°C of molecular replacement of Na2O by CaO, MgO, or Al2O3 in Na2O3SiO2 glass. (After S. English, J. Soc. Glass Technol. 8 (1924) 205. Reproduced with permission of the Society of Glass Technology.)

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Fig. 9.10 (A) Effect of the composition on the viscosity (Poise) of alkali borate melts. (B) Effect of the composition on the 1011 Pa s isokom in mixed-alkali borate glasses compared with that of single alkali borates. Curve marked V&S is the data of Visser and Stevels [14]. ((A) After T.J.M. Visser, J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 376; T.J.M. Visser, J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 395; T.J.M. Visser, J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 401. Reproduced with permission of Elsevier Science Publishers. (B) After C.M. Kuppinger, J.E. Shelby, J. Am. Ceram. Soc. 68 (1985) 463. Reproduced with permission of the American Ceramic Society.)

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appearance of the boron anomaly [15]. The apparent independence of the two phenomena is shown in Fig. 9.10B (after Kuppinger and Shelby [16]). It has been suggested [16] that the increase in viscosity is related to the replacement of the planar boroxol groups containing boron in three coordination, first, by three-dimensional tetraborate groups having boron in four coordination, and later, by nonbridging oxygens. The variation of viscosity with temperature in some compositions of commercial importance is shown in Fig. 9.11. Note that the soda lime

Fig. 9.11 Viscosity of some commercial silicate glasses. (Modified from R.H. Doremus, Glass Science, John Wiley and Sons, New York, 1973, p. 103. Reproduced with permission of the publisher.)

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silicate and the borosilicate glasses have similar annealing points, yet the latter has a much higher working point (and a higher melting point). The relative difficulty associated with the working and the melting of borosilicates has been used to call them “hard” glasses as opposed to “soft” glasses for the soda lime silicates and lead silicates. (The terminology has nothing to do with the hardness property discussed in Chapter 8.) Note also the curves for the soda lime and the alkali lead silicate glasses. The working points for both are around 1000°C; however, the lead silicates have a much lower softening point in comparison to that of soda lime silicates. The temperature interval between the working point and the softening point, called the working range is, hence, considerably larger for the alkali lead silicates. As a result, alkali lead silicate glasses are preferred for making intricate shapes. A glassblower or a machine can work on the lead silicates considerably longer before the need to reheat the workpiece arises. It happens that the thermal expansion coefficient of the alkali lead silicates is a close match to that of the soda lime silicates (see Chapter 10), a fact that is important in allowing direct sealing of intricate parts made out of the alkali lead silicates with simple shapes of inexpensive soda limes for many products. Household incandescent light bulbs are an excellent example where the outer shell is made of soda lime glass and the electrical lead mount structure is made of alkali lead silicate glass. Viscosity data for the chalcogenide systems are confined mostly to a measurement of the Tg. As expected, the presence of cross-linking atoms increases the Tg. By modifying the Gibbs-DiMarzio relation for the Tg of polymers, Sreeram et al. [17] were able to show that the Tg of Ge-Sb-AsSe-Te glasses could be written in the form: Tg ¼ Tg(0)/(1  β(  2)), where Tg(0) is the Tg of the parent Se-Te chain, is the average coordination number and β is a system constant. The term (  2) represents the average number of cross-linking bonds. Viscosity of heavy metal fluoride glasses has been studied extensively and is reviewed by Drexhage [18]. The Tg of most HMFG glasses of interest varies between 240°C and 340°C. The viscosity decreases rapidly on further heating till around 700–800°C, where their viscosity is of the order of only 0.05 Pa s.

9.5 Models for the temperature dependence of viscosity Several models have been published to describe the temperature dependence of liquid viscosity [19]. In this section, we focus on the most commonly used three-parameter models. We also introduce Angell’s concept of fragility

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[20], which describes the rate of viscosity change of a glass-forming liquid with temperature at the glass transition temperature (Tg).

9.5.1 Vogel-Fulcher-Tammann (VFT) equation In 1925, Gordon S. Fulcher introduced the following equation to describe the temperature dependence of the viscosity of glass-forming liquids [21]: log ηðT Þ ¼ A +

B : T  T0

(9.11)

Because of the large range of values, viscosity is traditionally presented on common logarithm basis (to the base 10) using the “log” notation as opposed to the “ln” notation for natural logarithm. Fulcher’s model was proposed empirically based on analysis of viscosity data for different silicate glassforming systems. Eq. (9.11) captures the temperature dependence of viscosity remarkably well, with only three fitting parameters: A, B, and T0. Eq. (9.11) is most commonly known as the Vogel-Fulcher-Tammann (VFT) equation, since the mathematical form was first proposed by Hans Vogel [22] in 1921 and subsequently studied by Tammann and Hesse [23] in 1926 in the context of organic liquids. Note that in the limit of T0 ! 0, Eq. (9.11) reduces to a simple Arrhenius relationship for viscosity, ηðT Þ ¼ η∞ exp ðΔH=RT Þ,

(9.12)

where ΔH is called the activation energy for viscous flow. The introduction of activation energy allows us to consider fluid motion to be associated with atomic mobility that increases with temperature. During the past century, the VFT equation has been the most frequently applied and successful model to describe the temperature dependence of viscosity. The VFT equation works well for a variety of liquids, most notably oxides, but it does not apply well for some other types of inorganic liquids and many organic liquids. Despite the notable success of the VFT equation, it often overpredicts viscosity values at low temperatures due to a mathematical divergence at T ¼ T0, where Eq. (9.11) predicts an infinite viscosity. In order to avoid the problem of divergence at finite temperature, several other viscosity equations have been proposed, including the Avramov-Milchev (AM) and Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equations described hereafter.

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9.5.2 Avramov-Milchev (AM) equation Whereas the VFT equation is purely empirical, the AM equation [24] is derived based on atomic hopping approach describing the kinetics of molecular motion in supercooled liquids. The main assumption of the model is that the structural disorder leads to a random probability distribution of activation energies for molecular transport. The distribution of activation energies is expressed in terms of the entropy of the system. The viscosity of the system is described in terms of the average transition rate of a hopping particle, calculated from this probability distribution function. The resulting AM equation is  τ α logη ¼ log η∞ + , (9.13) T where η∞, τ, and α are treated as fitting parameters. The model predicts a non-Arrhenius dependence of viscosity on temperature, which can successfully reproduce experimental viscosity data. The AM equation is able to describe molecular transport properly in both the crystalline and the glassy state. Unlike the VFT equation, the AM model does not predict a divergence of viscosity at any finite temperature, and as a result is more accurate than VFT at low temperatures.

9.5.3 Angell’s concept of fragility The Angell plot describing the temperature dependence of viscosity is one of the most important figures in all of glass science. As shown in Fig. 9.12, Angell plotted the base-10 logarithm of viscosity, log η, as a function of the Tg-scaled inverse temperature, Tg/T [20]. The glass transition temperature, Tg, is defined by Angell as the temperature at which the liquid viscosity equals 1012 Pa s, and T is the absolute temperature expressed in Kelvin. The slope of the Angell curve at Tg defines the fragility index m, ∂ log η  : (9.14) m  ∂ Tg =T T ¼Tg

The fragility index describes the rate of change in the liquid dynamics on cooling through the glass transition. As shown in Fig. 9.12, different glassforming liquids exhibit variation in the degree of non-Arrhenius scaling. Based on this figure, Angell classified liquids as either “strong” or “fragile” depending on whether they exhibit an Arrhenius or non-Arrhenius scaling of viscosity with temperature, respectively. The Arrhenius “strong”

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14 12 12

STRONG

8

GeO2 Si O

2

6 3 ⋅2

6

l2 O

SiO2 iO 2

3 +P

2S

2

K+Ca2+NO3−

Cl 2

Zn

0

0

y+

+P Cl 2

K+Bi3+Cl−

−2

–2

–4

− Cl

Zn

m, o-Xylene ethanol

o-Terphenyl Toluene

0

0.2

2

CH

O⋅ 2 Na

y+ Cl −

O⋅ A

4

Ca

4

log (viscosity in Pa s)

8 log (viscosity in poise)

10

As2S3

10

FRAGILE

m, o-Fluorotoluene Chlorobenzene

0.4

0.6

0.8

−4 1.0

Tg/T

Fig. 9.12 Angell diagram of log(viscosity) versus Tg/T for “strong” and “fragile” liquids. (Modified from C.A. Angell, J. Non-Cryst. Solids 73 (1985) 1; C.A. Angell, J. Non-Cryst. Solids 102 (1988) 205. Reproduced with permission of Elsevier Science Publishers.)

liquids normally exhibit small property changes during glass transition since they have stable structures with a high degree of short-range order. On the other hand, non-Arrhenius “fragile” liquids exhibit dramatic changes in properties in the glass transition range reflective of a rapidly changing activation barrier to viscous flow.

9.5.4 Entropy model of viscosity: The Adam-Gibbs equation Adam and Gibbs [25] derived an entropy-based theory of structural relaxation that describes the viscosity-temperature relationship of a liquid. The main assumption of the Adam-Gibbs model is that a liquid consists of a number of regions or subsystems that can cooperatively rearrange. Each region is composed of a group of molecules or monomers that can rearrange itself independently of its surroundings. As the liquid is cooled, the size of the cooperatively

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rearranging regions grows progressively larger. This yields a decrease in the configurational entropy of the system, which in turn leads to an increase in viscosity. This continues until the configurational entropy becomes zero at low temperature. The system consists of just one cooperatively relaxing region when the configurational entropy vanishes. At this point, there is no further freedom for the system to rearrange its structure, and the molecular relaxation time (i.e., viscosity) of the system becomes infinite. In the Adam-Gibbs formulation, the molecular relaxation time (viscosity) is inversely related to the configurational entropy of the system. Thus, the viscosity of a given composition x can be computed as a function of its configurational entropy logηðT Þ ¼ logη∞ +

B , TSc ðT Þ

(9.15)

where Sc(T) is the temperature-dependent configurational entropy of the liquid and B is an effective activation barrier. The Adam-Gibbs model linking the viscosity of a glass-forming liquid with its configurational entropy has met with remarkable success in describing the relaxation behavior of a wide variety of systems [26]. However, it does not provide a means for calculating the size of the cooperatively rearranging regions or the constant B, which is left as a fitting parameter. The Adam-Gibbs model can provide some physical insights regarding Angell’s classification of strong and fragile liquids. Due to the structural differences between strong and fragile liquids, fragile liquids are expected to have large change in configurational entropy with temperature, leading to a non-Arrhenius scaling of viscosity according to Eq. (9.14). On the other hand, strong liquids are expected to have a fairly constant configurational entropy, yielding an Arrhenius dependence of viscosity as in Eq. (9.12).

9.5.5 Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equation The recent MYEGA model of viscosity [27] is derived from the AdamGibbs equation described earlier, where the configurational entropy, Sc(T,x), is expressed by a Boltzmann distribution, leading to   K C logηðT Þ ¼ logη∞ + exp (9.16) T T where η∞, K, and C are three fitting parameters. The MYEGA model provides improved fitting of viscosity-temperature curves, building on the entropy model of Adam and Gibbs.

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Fig. 9.13 (A) Low-temperature extrapolation test, where the viscosity models are fit to high-temperature viscosity data and then extrapolated to predict the 1011 Pa s isokom temperature. (Here, “Current Model” means the MYEGA model.) (B) Error in the predicted 1011 Pa s isokom for 85 different compositions. The compositions on the horizontal axis are ordered in terms of descending error for the three models. A given position on the horizontal axis generally corresponds to three different liquids. (C) Root-mean-square error in the predicted isokom temperature using the three different models. (D) Average error in the predicted isokom temperature. (Modified from J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Proc. Natl. Acad. Sci. U. S. A. 106 (2009) 19780.)

The intrinsic differences among the VFT, AM, and MYEGA models have been studied in detail in Ref. [27]. Fig. 9.13 shows the results of a model extrapolation test of the three viscosity models. Here, each of the three viscosity models is independently fitted to viscosity versus temperature data at high temperatures and then extrapolated to low temperatures to predict the 1011 Pa s isokom point. The AM equation systematically underpredicts the 1011 Pa s isokom temperature, whereas VFT overpredicts the same isokom. In contrast, the MYEGA model exhibits no systematic error when performing this low-temperature extrapolation. The systematic error of AM

The viscosity of glass

239

is due to an unphysical divergence of configurational entropy in the hightemperature limit. In contrast, the error of VFT is a result of the divergence of viscosity at low temperatures.

9.6 Composition dependence of viscosity The VFT, AM, and MYEGA models of viscosity can all be rewritten in terms of three common parameters: (i) the glass transition temperature, Tg; (ii) the fragility, m; and (iii) the extrapolated infinite temperature viscosity, η∞. With these definitions, the VFT expression of Eq. (9.11) becomes ð12  logη∞ Þ2   logηðT Þ ¼ logη∞ + , m T =Tg  1 + ð12  log η∞ Þ

(9.17)

the AM expression of Eq. (9.13) becomes  m=ð12 logη∞ Þ Tg logηðT Þ ¼ logη∞ + ð12  logη∞ Þ , T

(9.18)

and the MYEGA model in Eq. (9.16) can be rewritten as

   Tg Tg m 1 : 1 log ηðT Þ ¼ log η∞ + ð12  log η∞ Þ exp T T 12  log η∞ (9.19) Empirically, it has been found that the infinite temperature limit of viscosity is constant (i.e., independent of composition) for a given model (η∞ ¼ 103.87 Pa s using the VFT equation, η∞ ¼ 101.74 Pa s following the AM model, or η∞ ¼ 102.93 Pa s using the MYEGA equation). The difference in the specific values of η∞ is reflective of intrinsic differences in the shapes of the three viscosity equations (9.17)–(9.19) and the fact that η∞ can never be directly measured, since it corresponds to the extrapolated viscosity at infinite temperature. Since η∞ is a constant, all of the composition dependence of the viscosity in Eqs. (9.17)–(9.19) can be captured by changes in the remaining two parameters: the glass transition temperature (Tg) and the fragility (m). Gupta and Mauro [28, 29] developed an approach for modeling the composition dependence of Tg and m known as temperature-dependent constraint theory. This is an extension of the Phillips-Thorpe constraint theory (discussed in Section 3.1.2) incorporating the effect of temperature on the rigidity of the glass-forming network. Following temperature-dependent constraint

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theory, the composition (x) dependence of Tg can be calculated as the following:



d  n Tg ðxR Þ, xR Tg ðxÞ f Tg ðxR Þ, xR

¼

, ¼ (9.20) Tg ðxR Þ f Tg ðxÞ, x d  n Tg ðxÞ, x where xR is a reference composition with a known value of Tg, d ¼ 3 is the dimensionality of the network, f(T,x) is the number of atomic degrees of freedom for composition x at temperature T, and n(T,x) is the average number of constraints per atom. A greater number of rigid constraints in the liquid network leads to a higher glass transition temperature, that is, the entire viscosity curve is shifted to higher temperatures for a more rigid network. The composition dependence of fragility can be calculated by   ∂ ln f ðT , xÞ mðxÞ ¼ m0 1 + , (9.21) ∂ ln T where m0 ¼ 12 – log η∞ is the fragility of a perfectly strong liquid. Note that the glass transition temperature is directly calculated from the number of constraints per atom, while the fragility is calculated from the temperature derivative of atomic degrees of freedom. Temperature-dependent constraint theory has been applied to predict Tg and m of various series of glass systems, covering borates [29], borosilicates [30], phosphates [31, 32], and borophosphate glasses [33]. The temperature-dependent constraint models are developed by considering the structural and topological role of each species in the network. Fig. 9.14 shows model calculations of Tg for sodium borosilicate and soda lime borosilicate systems. The constraint model offers simple calculations that enable exploration of new composition spaces in an economical and effective way. Topological constraint modeling is discussed in greater detail in Chapter 21.

9.7 Viscosity of cathedral glass at room temperature (the “urban legend”) It is not so uncommon a belief that glass in medieval cathedral windowpanes has become thicker at the bottom due to flow under its own weight over centuries and, hence, glass must be a (supercooled) “liquid” even at room temperature as opposed to being a “solid.” This urban legend seems to have started as early as 1894 and has been cited even in recent years. Some historical accounts of the many associated observations have been provided by Preston [34] and Newton [35]. Assuming that glass at room temperature

The viscosity of glass

241

Fig. 9.14 Model calculations of Tg as a function of composition using temperature-dependent constraint theory for (A) sodium borosilicate and (B) soda lime borosilicate systems. Modified from Smedskjaer et al. [30].

behaves as if it were a liquid at its fictive temperature (Section 2.2) of around 816 K, Zanotto and Gupta [36] have estimated the viscosity of glass to be around 1041 Pa s, yielding an estimate of time for any dimensional variations in glass to be around 1023 years. As shown in Fig. 9.15, more recent measurements [37] of the viscosity of a medieval cathedral glass composition used in Westminster Abbey have placed the room temperature viscosity around 1024–1025 Pa s, many orders of magnitude lower than previous estimates but still too high to observe measurable flow on a human timescale. It is almost certain that glass windowpanes in old cathedrals were nonuniform in thickness to begin with as a result of the glass-forming technique used, and thicker at the bottom because of an intentional or unintentional preference to mounting them that way.

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Fig. 9.15 Viscosity and flow rate of cathedral glass at room temperature as a function of fictive temperature. (Modified from O. Gulbiten, J.C. Mauro, X. Guo, O.N. Boratav, J. Am. Ceram. Soc. 101 (2018) 5.)

9.8 Non-Newtonian viscosity Any fluid where η changes with the shear-strain rate e_ xy in Eq. (9.1) is called a non-Newtonian fluid. Common classifications of non-Newtonian fluids are discussed in the following list. Their shear stress (σ xy) versus shear-strain rates (_e xy ) and η versus e_ xy are schematically shown in Fig. 9.16. (i) Bingham plastic: This material shows no deformation with time until it exceeds a yield point. Thereafter, the relationship between the shear stress and the shear-strain rate is linear. (Some paints, slurries may approximate this model over certain range of temperatures. Toothpaste is another example of a Bingham plastic.) (ii) Pseudoplastic (shear thinning): The apparent viscosity decreases with the magnitude of the shear stress. Glass at high shearing stresses may approach this behavior. (iii) Dilatancy (shear thickening): This is the inverse of the pseudoplastic behavior. The apparent viscosity increases with increasing shear stress. Concentrated ceramic slips often show dilatant behavior. (iv) Thixotropy: The material shows a reversible decrease in the apparent viscosity with time at a constant shear rate. Unlike shear thinning,

The viscosity of glass

243

Fig. 9.16 Classes of non-Newtonian behavior.

the viscosity depends on time, and not on the magnitude of the shear stress. Thixotropy occurs because of a reversible structural change with time. Hysteresis curves develop (Fig. 9.17); the apparent viscosity decreasing and approaching a limiting value with repeated cycling. Pseudoplastic or shear thinning behavior is of considerable interest in glass forming. During high-speed pressing operations, glass may have sufficiently high shearing rates to cause departure from the Newtonian behavior (in addition to lowered viscosity from localized heating). Shear thinning may lead the glass to flow out of a controlled shape during the forming

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Fig. 9.17 Thixotropy. Table 9.2 Shear-strain rates marking the onset of shear Inverse relaxation Temperature Newtonian time (s21) (°C) viscosity (poise)

530 570 630 690 720 800 900

1014 1012 1010 108 107 106 105

5  103 4  101 2  101 1  102 1  103 1  104 1  105

thinning Critical strain rate (s21) for 10% decrease from Newtonian viscosity

5  106 2  104 6  103 2  101 1.2  100 7  100 4  101

After J. H. Simmons, C. J. Simmons, Am. Ceram. Soc. Bull. 68 (1989) 1949.

process, particularly during high-speed pressing and blowing operations. In addition, the lowered viscosity could also cause increased ionic mobility (Stokes-Einstein relationship) leading to increased phase separation which, in turn, would cause workability problems during glass forming. Much of the research in the area of non-Newtonian viscosity of glass has been performed by Br€ uckner and coworkers [38, 39] and Simmons and Simmons [40]. Table 9.2, taken from Simmons and Simmons, shows that the onset of non-Newtonian behavior in a soda lime glass may occur at as low as 5  106 s1 strain rate at 530°C, whereas the viscosity remains Newtonian up to a strain rate of 40 s1 at 900°C. A fit of the normalized viscosity η/η0 (where η0 ¼ the Newtonian viscosity) to the equation: η 1 ¼ η0 1 + e_ xy η0 =σ ∞

(9.22)

The viscosity of glass

245

Fig. 9.18 Normalized viscosity versus normalized strain rate at various temperatures and for various methods. (After J.H. Simmons, C.J. Simmons, Am. Ceram. Soc. Bull. 68 (1989) 1949. Reproduced with permission of the American Ceramic Society.)

where σ ∞ is the high-temperature cohesive shear strength, for three different types of experiments is shown in Fig. 9.18. The estimated values of σ ∞ are plotted in Fig. 9.19. It may be noted that σ ∞ increases steadily with the viscosity. Conceptually, cohesive shear strength should be the breaking strength of the SiO network-forming bond. As Br€ uckner [38] points out, not only does the viscosity become nonlinear at high shearing rates, the shear stress also builds up rapidly as a result of inadequate rate of stress relaxation. If the shear stress exceeds the cohesive shear strength σ ∞, then it is possible to have fracture in a “seemingly” liquid state (see Exercise 2.1 regarding the distinction between a solid and a liquid). Thus, high-speed pressing and blowing processes require a consideration of non-Newtonian flows and high-temperature fracture of glass. The concept of “liquid fracture,” or even high deformations before fracture in seemingly liquid glass network is important to our thoughts about the strength of solid glass discussed in Chapter 18. Finally, one may argue that a material’s response to shear is controlled by its plastic yield strength, its cohesive shear strength, its shear modulus, and the applied shearing rate. Application of shear at ordinary temperatures causes the material to behave solid-like: it deforms elastically and plastically if the applied shearing stress exceeds the yield strength. Fracture can occur if

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Fig. 9.19 Variation of the cohesive strength versus Newtonian viscosity. (After J.H. Simmons, C.J. Simmons, Am. Ceram. Soc. Bull. 68 (1989) 1949. Reproduced with permission of the American Ceramic Society.)

the magnitude of the shear stress exceeds the cohesive strength. As temperature increases, the yield strength decreases sufficiently to allow flow to occur, and the material is appropriately termed a “liquid.” Because of flow, shear stress relaxes to remain below the fracture strength. However, at high shearing rates, even a liquid could fracture.

9.9 Volume viscosity Thus far, we discussed shear-associated viscosity which leads to a shape change by a rearrangement of the interstitial volume, but not to a change in volume itself. The volume viscosity ηv is defined by

ðp  p0 Þ ¼ ηv V ,

(9.23)

where p0 is the pressure necessary to maintain the material at the same volume. The equation is obtained by replacing shear stress by (p  p0) and shear strain rate by volume strain rate in Eq. (9.1). Volume viscosity influences the interstitial volume presumably by the changing of bond angles and a consequent cooperative readjustment of the packing. The concept of volume viscosity has been discussed in relation to the isothermal relaxation of volume during glass transition by Davies and Jones [41], who suggested that the kinetics of glass transition involved a relaxation time which was related

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247

to, more appropriately, the volume viscosity and not the shear viscosity. (The influence of viscosity on glass transition is discussed in more detail in Chapter 13.) Davies and Jones further showed that, at least in the glass transition range, the activation energies for the shear viscosity and the volume viscosity were about the same (for more discussion, see Mazurin [42]). The same may not be true below glass transition temperatures. For instance, Mizouchi and Cooper [43] showed that the rates of density recovery at room temperature in pressure-densified B2O3 were measurable and depended on the density as well as on the extent of initial densification (Section 7.3). Thus, whereas the shear viscosity at room temperature is effectively infinite, by contrast, the volume fluidity at low temperatures appears to be significantly lower. Likewise, volume viscosity may be a major factor determining stress relaxation during the chemical strengthening of glass through ion exchange. Unfortunately, our understanding of the exact mechanisms underlying the concept of the volume viscosity at temperatures well below the glass transition range is not satisfactory.

9.10 Glass greats: G.S. Fulcher and R.W. Douglas Gordon Scott Fulcher (1884–1971, Fig. 9.20) was a pioneer in the analytical study of viscosity, as well as one of the principal inventors of electrocast ceramic refractories. His 1925 paper, in which he proposed the VFT equation for viscosity [21], is one of the most influential papers ever published in the field of glass science and the second most highly cited paper ever in the Journal of the American Ceramic Society. Fulcher’s influence on the scientific community is much broader than just his work in glass and ceramics. As managing editor of Physical Review in the 1920s, Fulcher proposed the

Fig. 9.20 Gordon Scott Fulcher (1884–1971), developer of the VFT equation for supercooled liquid viscosity.

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Fundamentals of Inorganic Glasses

Fig. 9.21 Ronald Walter Douglas (1910–2000), pioneer in the measurement of viscosity at high temperature.

modern system of abstracting that is now used by all scientific publications. Fulcher also published two books on metacognition (“thinking about thinking”): Better Thinking for Better Living and Common Sense Decision-Making. A detailed biography of G.S. Fulcher can be found in Ref. [44]. Ronald Walter Douglas (1910–2000, Fig. 9.21), the youngest of eight children, grew up to become one of the most influential pioneers of modern glass technology in the United Kingdom. As a researcher at General Electric, Douglas developed methods for measuring the viscosity of glass at high temperatures that are still in use today. He later joined the faculty at Sheffield University as Professor of Glass Technology, a position he held for two decades (1955–1975). He was one of the most influential figures in the Society of Glass Technology for over 40 years. Professor Douglas taught the first formal lessons on glass science to the senior author of this book (AKV).

Summary Viscosity is a liquid-state property. The mass can be thought of as a plastic substance with zero yield strength. The time rate of deformation (flow rate) is proportional to the applied shear stress; the constant of proportionality is called (Newtonian) viscosity (units: poise or P in cgs and Pa s in SI; 1 Pa s ¼ 10 P). With the application of heat, glass gradually softens and converts to a liquid. Viscosity commonly ranges between 1016 Pa s (solid-like)

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249

and 1 Pa s (liquid-like). The temperature dependence is described by several different models, including the VFT, AM, and MYEGA equations. The Adam-Gibbs model considers viscous flow as originating from cooperatively rearranging regions in the liquid. Glass-forming liquids can be classified as either “strong” or “fragile”; the more fragile liquids deviate more from an Arrhenius-type temperature dependence of viscosity because of the availability of large number of configurations. For a commercial soda lime silicate glass, the working point (104 P) is 1000° C; the softening point (107.65 P) is 750° C; the annealing point (1013 P) is 530° C; and the strain point (1014.5 P) is 480° C. Viscosity is measured by the falling sphere or rotating cylinder methods in the liquid range; parallel plate, fiber elongation, and beam bending methods in the higher viscosity ranges. When high shearing rates are applied, it is possible for the viscosity of molten glass to become non-Newtonian; for example, shear thinning may occur. A network fracture may occur even in seemingly liquid state.

Online Resources (1) Glass Viscosity Standards Online: http://glassproperties.com/ standards/. (2) Viscosity Demonstration by T.J. Kiczenski: https://www.youtube. com/watch?v¼YArf6UCyHA4. (3) “The Viscosity of Glass-Forming Oxide Liquids” by Edgar Zanotto: https://www.youtube.com/watch?v¼pGTMnFKyfuQ.

Exercises (1) A fiber of uniform radius r and length L is suspended in a furnace holding a constant temperature. A mass M (much greater than the mass of the fiber) is attached to the bottom of the fiber. Develop an expression to determine the viscosity using the measured elongation rate. (2) Develop the formula for the computation of viscosity by measuring the sag rate at mid-point in a three-point beam bending method. (3) Develop the formula for the standardized measurement of the softening point with and without the surface tension correction. (4) Calculate the rate of extension under its own weight of a clad-core composite glass fiber of length L, having radii r1 (for the core) and r2 (for the clad), and suspended in a uniform temperature furnace such

250

(5)

(6)

(7)

(8)

Fundamentals of Inorganic Glasses

that the corresponding viscosities are η1 and η2, respectively. Hence, estimate the change in the extension rate when (r2  r1)/r2  0.1 and η2 ¼ 100 η1. [Note: Because of alkali volatilization and surface silica enrichment, the high “skin” viscosity is typical, and affects many glass-forming processes.] Calculate the rate of rise of spherical bubbles of different sizes in a molten glass tank having glass of density 2.5 g/cm3 and viscosity 10 Pa s. Show that micron-sized bubbles may take months to rise from the tank bottom to the surface 125 cm above. [Hence, fine bubbles in glass tanks, otherwise called “seeds,” must be removed generally by chemical means.] Two parallel glass plates measuring 10  20  0.03 cm have been fritsealed at all four edges such that there is a 0.0015 cm space in between. Assume that a hole in one of the corners allows air pressure to be maintained atmospheric. How much will the upper plate sag in the middle if the assembly is allowed to heat at its annealing point for 15 min? Develop an approximate expression for the extension rate, under its own weight, of a glass plate of length l, width w, thickness t, suspended vertically and viscosity η. Hence, estimate the value of η for the glass to extend 1 mm at room temperature. In a biaxial elastic flexure test, a plate specimen of thickness t is supported on a ring of radius a and is loaded centrally by a force F using a ball. In the limit of no overhang, the deflection is given by δ ¼ ð3 + νÞFa2 =½16π ð1 + νÞD , where D is called the flexural modulus given by   D ¼ Et3 = 12 1  ν2 Develop an expression relating the sag rate to the viscosity of the specimen.

References [1] [2] [3] [4] [5] [6] [7] [8]

J.N. Goodier, Phil. Mag. 22 (1936) 678. F.T. Trouton, Proc. R. Soc. London A77 (1906) 426. R.W. Douglas, J. Soc. Glas. Technol. 31 (1947) 74. J.T. Littleton, J. Soc. Glas. Technol. 24 (1940) 176T. H.R. Lillie, Phys. Rev. 36 (1930) 347. H.R. Lillie, J. Amer. Ceram. Soc. 12 (1929) 516. G.F.C. Searle, Proc. Camb. Philos. Soc. (1912). A. Napolitano, P.B. Macedo, E.G. Hawkins, J. Res. Nat. Bur. Standards 69A (1965) 449.

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[9] H. Rawson, Properties and Applications of Glass, Elsevier, 1980, p. 44. [10] G. Cseh, N.Q. Chinh, P. Tasnadi, P. Szommer, A. Juhasz, J. Mater. Sci. 32 (1997) 1733. [11] R.W. Douglas, W.L. Armstrong, J.P. Edward, D. Hall, Glass Technol. 6 (1965) 52. [12] A.K. Varshneya, N.H. Burlingame, W.H. Schultze, Glastech. Ber. 63K (1990) 447. [13] (a)R.F. Bartholomew, M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Technol., vol. 22, Academic Press, 1982, pp. 75–127; (b)J. Deubener, R. M€ uller, H. Behrens, G. Heide, J. Non-Cryst. Solids, 330 (2003) 268. [14] T.J.M. Visser, J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 376; T.J.M. Visser, J. M. Stevels, J. Non-Cryst. Solids 7 (1972) 395; T.J.M. Visser, J.M. Stevels, J. Non-Cryst. Solids 7 (1972) 401. [15] S.V. Nemilov, J. Appl. Chem. USSR Eng. Trans. 42 (1969) 46. [16] C.M. Kuppinger, J.E. Shelby, J. Am. Ceram. Soc. 68 (1985) 463. [17] A.N. Sreeram, D.R. Swiler, A.K. Varshneya, J. Non-Cryst. Solids 127 (1991) 287. [18] M.G. Drexhage, M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Technol., vol. 26, Glass IV, Academic Press, New York, 1985, , pp. 151–243. [19] Q. Zheng, J.C. Mauro, J. Am. Ceram. Soc. 100 (2017) 6. [20] C.A. Angell, J. Non-Cryst. Solids 73 (1985) 1. C.A. Angell, J. Non-Cryst. Solids 102 (1988) 205. [21] G.S. Fulcher, J. Am. Ceram. Soc. 8 (1925) 339. [22] H. Vogel, Phys. Z. 22 (1921) 645. [23] G. Tammann, W. Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245. [24] I. Avramov, A. Milchev, J. Non-Cryst. Solids 104 (1988) 253. [25] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [26] G.W. Scherer, J. Am. Ceram. Soc. 67 (1984) 504. [27] J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Proc. Natl. Acad. Sci. U. S. A. 106 (2009) 19780. [28] P.K. Gupta, J.C. Mauro, J. Chem. Phys. 130 (2009) 094503. [29] J.C. Mauro, P.K. Gupta, R.J. Loucks, J. Chem. Phys. 130 (2009) 234503. [30] M.M. Smedskjaer, J.C. Mauro, R.E. Youngman, C.L. Hogue, M. Potuzak, Y. Yue, J. Phys. Chem. B 115 (2011) 12930. [31] C. Hermansen, J.C. Mauro, Y. Yue, J. Chem. Phys. 140 (2014) 154501. [32] C. Hermansen, B.P. Rodrigues, L. Wondraczek, Y. Yue, J. Chem. Phys. 141 (2014) 244502. [33] C. Hermansen, R.E. Youngman, J. Wang, Y. Yue, J. Chem. Phys. 142 (2015) 184503. [34] F.W. Preston, Glass Technol. 14 (1973) 20. [35] R.G. Newton, Glass Technol. 37 (1996) 143. [36] E.D. Zanotto, P.K. Gupta, Am. J. Phys. 67 (1999) 260. [37] O. Gulbiten, J.C. Mauro, X. Guo, O.N. Boratav, J. Am. Ceram. Soc. 101 (2018) 5. [38] R. Br€ uckner, Ceram. Trans. 29 (1992) 3. [39] P. Manns, R. Br€ uckner, Glastech. Ber. 61 (1988) 46. [40] J.H. Simmons, C.J. Simmons, Am. Ceram. Soc. Bull. 68 (1989) 1949. [41] R.O. Davies, G.O. Jones, Adv. Phys. 2 (1953) 370. [42] O.V. Mazurin, J. Non-Cryst. Solids 87 (1986) 392. [43] N. Mizouchi, A.R. Cooper, J. Am. Ceram. Soc. 56 (1973) 320. [44] J.C. Mauro, Front. Mater. 1 (2014) 25.