Response and structure of glass melts under extreme forming processes

Response and structure of glass melts under extreme forming processes

Journal of Non-Crystalline Solids 73 (1985) 421-449 North-Holland, Amsterdam 421 R E S P O N S E AND S T R U C T U R E OF GLASS M E L T S UNDER EXTR...

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Journal of Non-Crystalline Solids 73 (1985) 421-449 North-Holland, Amsterdam


R E S P O N S E AND S T R U C T U R E OF GLASS M E L T S UNDER EXTREME FORMING PROCESSES Rolf B R O C K N E R lnstitut fur Nichtmetallische Werkstoffe, Anorganische Werkstoffe, Technische Universiti~t Berlin, Germany There is a practical and scientific interest in the behaviour and structure of glass melts under deformation conditions leading up to the limit of material strength. The following topics are treated in a compressed manner under this aspect: flow behaviour and high-temperature fracture (Newtonian flow, non-Newtonian flow, viscoelastic behaviour and fracture); anisotropy and orientation phenomena (birefringence in phase-separated and single-phase glasses, flow birefringence). An outlook is given on problems of this topic still being unsolved and still being of interest to be treated during the next 20 years.

1. Introduction One of the developments of modern glass technology for the so-called consumer glass types may be summarized in the statement, that the production rates are rising permanently. There are limitations set due to the properties of the forming materials and of the glass melts. The main limits are heat flow, high temperature strength, and flow behaviour. This is valid not only for the usual consumer glass types but also for some very new categories of glasses as metal glasses, halide glasses and chalcogenide glasses; in total a large field, which will be expanded by structural aspects. Nevertheless, within the scope of this paper we like to take a first step in that direction, hoping that we shall know a lot more about these problems at Norbert Kreidl's 100th birthday. Thus, I like to exclude new forming materials and to limit this paper to commercial consumer glass types and in a few cases to some phosphate and sulphide glasses.

2. Flow behaviour and high-temperature fracture One of the most important properties of glass melts is the flow behaviour. This is true not only for the melting and forming process in the glass production industries but also for processes within our earth globe in the form of volcanic and magmatic activities. However, from a structural viewpoint the flow behaviour is very important and interesting too. 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


R. Br~ckner / Glass melts under extreme forming processes

I 20 / x 16 ~/~,'~ lower annealing / / / S ~ "~ temperature c

/ / / ) d k te.,,ro~u~ /glass


//.-'//~]~'~ffg~x"~,,L ///~I] ~ I _ ~ . ~ .

me|ring and

--Temperature in *C



Fig. 1. Viscosity-temperature curve and various characteristic sections and points for glass melts.

2.1. Newtonian flow behaviour The most amazing fact of high viscous flow behaviour of oxide glass melts is, that it is usually a purely Newtonian one, that means, it can be characterized by only one material property, the Newtonian viscosity. The normal viscosity-temperature behaviour is shown schematically in fig. 1. Many orders of magnitude are covered, from viscosity of the stable melt, where the viscosity is of the order of 10 ] to 102 dPa s down to the glass transition temperature where usually a viscosity of 1013 dPa s is obtained. At higher viscosities a deviation from the usual viscosity temperature curve takes place, because the time for producing the metastable equilibrium of the high viscous melt becomes too long, and the viscosity becomes time-dependent as is shown in fig. 2. The equilibrium viscosity is obtained from two directions. The upper curves in this figure were measured with glass samples, which have been brought into a metastable equilibrium with respect to a temperature - the so-called "fictive temperature" - which is lower than the temperature at which the viscosity measurement was performed. The lower curves in this figure were measured with glass samples, which have been brought into a metastable equilibrium with respect to a fictive temperature, which is higher than the temperature at which the viscosity was measured [1] (see also ref. [2] for lower viscosities). From a structural view point one may state that the silicate network or the network fragments, respectively, need time to arrange themselves into energetically favourable positions during the viscosity measurement. Values determined as equilibrium viscosities are Newtonian ones. Starting from the viscosity temperature curves of the glass-forming oxides B203, GeO 2, and SiO2, it is very typical for the GeO 2 and SiO 2 melts, that curves will shift to lower temperatures with increasing incorporation of network modifier oxides like Na20, etc. [3-5]. The reason is, that the continuous network will be split into smaller fragments, causing a decrease of the viscosity. On the other hand, the boron oxide melt will become more viscous with increasing alkali oxide [6]. The reason of that is a drastic change in the structure of the boron oxide melt and glass. The alkali oxide does not produce non-bridging oxygens as in the case of the silicate or germanate melts, but

R. Brf~ckner / Glass melts under extreme forming processes ~>




14 5

~s.51 14 0


! l


I o





lime (min)


145 ,..~..o-o~o_~"






I _



14 5-



^~x~x..~. | \EquihbriumViscosity x....~, 2 60,10 I~ dPos


I 140



I iooo lime (rain)

140 01

lime -i


// 2000 --'-



Fig. 2. Time-dependent viscosity for glass samples with various fictive temperatures Tr (higher and lower fictive temperatures than the temperatures T, at which the viscosity was measured). The upper curves are obtained for sampels with Tr < T,, the lower curves for samples with Tr > 7~ (Lillie, 1933).

causes the boron atoms to change from a threefold to a fourfold coordination. Thus, an increasing three-dimensional connected network will be produced from a two-dimensional connected network; that means more and more BO4 groups are formed from BO 3 groups by the addition of alkali oxide as was shown by nuclear magnetic resonance investigations by Bray et al. [7-9] and the viscosity temperature curve will shift to higher temperatures. The pronounced higher viscosity curve of the SiO 2 melt as compared with that of the GeO 2 melt is the result of the bonding strength between Si and O greater than that between Ge and O; this is the most significant difference between these two substances. The addition of 15 mol.% N a 2 0 to the SiO 2 network and the influence on viscosity of those melts causes a drastic decrease of over 10 orders of magnitude, followed by a smaller decrease with further alkali oxide content [3]. The incorporation of alkali oxide produces non-bridging oxygen atoms, and an increase of the portion of ionic bonding type of surrounding bridging oxygen atoms. Hence, a shift to a more ionic bonding of the bridging oxygens will result when N a 2 0 is added to SiO 2 as could be shown from electron spectroscopy [10,11]. Another example may be given with alkali-aluminosilicate melts. It is well known from feldspar structures that aluminum can substitude for silicon, if the electroneutrality is maintained by the charge of an alkali ion [12,13] (for further reference see [13]). Extended measurements on viscosity in these systems and in those with alkaline earth have been made [14]. The increase of viscosity may be explained by the reduction of the concentration of non-bridging oxygens and therefore by a more stable network. But the viscosity is also


R. Bri~ckner / Glass melts under extreme forming processes

increased, when SiO 2 is replaced by A1203 without changing the N a 2 0 content [15]. Here, two non-bridging oxygens are replaced by one bridging oxygen atom. In that connection the coordination behaviour of aluminum is of special interest. The old opinion, that aluminum is fourfold coordinated for ratios A1/Na < 1 and > 1 sixfold coordinated, is not strictly tenable. It is concluded, that for ratios < 1 and > 1 besides fourfold also sixfold coordinated aluminum will be present in the glass. However, in the melt at very high temperatures only fourfold coordinated aluminum will exist below and above the ratio A 1 / N a [13,16]. As a conclusion from the relation between viscosity and structure one may state the viscosity can give information on some structural questions as the formation of bridging and non-bridging oxygens, the interaction energy of the structural or flow units of the melt, and the coordination tendency of incorporated cations. It is even possible to measure the different influence of noblegas-like and noble-gas-unlike cations on the flow resistance, which is usually connected with the structural strength of a melt. Thus, for example, the substitution of the noble-gas-like strontium by the noble-gas-unlike lead atoms causes a much easier thermally activated change of positions due to the higher polarizability, and therefore a much lower viscosity is the result (of course at comparable other conditions) [17]. However, the lack of today's knowledge is, that the connection between viscosity on the one hand and melt composition and structure on the other, is only a very qualitative one, because the flow behaviour on the atomic scale is a property of a strong interaction process.

2.2. Viscoelastic behaviour and fracture The above mentioned examples have given some insight into the usefulness of viscosity measurements of silicate melts for structural information. However, if one compares viscosity with the stress-strain relations for industrial deformation p~ocesses at different strain rates, remarkable differences will arise. Figure 3 shows on the left the stress-strain rate relation for a typical viscosity measurement. A linear variation with the time of the deformation produces a constant deformation rate "i' and a constant stress a. The ratio a / ? is the well-known Newtonian viscosity. The situation is different, if a sudden deformation of a viscous material is produced as is shown on the right of fig. 3. This situation is still idealized by a constant strain rate during the deformation process. After the maximum stress is attained the material relaxes. This deformation process cannot be described by the usual viscosity. Therefore the theory of viscoelasticity introduces some new concepts, such as the stress relaxation modulus [18,19] as defined in the equations [20] of fig. 4. The stress relaxation modulus is a function of time (t), temperature (T), and composition of a melt. The time integrated product of this modulus and the strain rate gives a strain rate-dependent maximum for the induced stress, followed by a

R. Br~ckner / Glass melts under extreme forming processes

Viscosity Measurement


Actuol Oefomotion Behoviour









O(t) O(t)T m-




Fig. 3. C o m p a r i s o n between the s t r e s s - s t r a i n - s t r a i n - r a t e relation for a viscosity m e a s u r e m e n t and for an actual d e f o r m a t i o n process.

relaxation process. If the deformation and its rate are high enough, so that the induced stress exceeds the strength of the material, a fract',lre will occur within the viscous liquid material, which would not have been fractured at lower strain rates. This behaviour is called high-temperature fracture above T~. At high deformation rates the elastic response dominates the viscous response of the material. The higher the deformation rate, the more elastic will be the theological response of the material. Thus, a viscous silicate melt can behave as an elastic and brittle body for which the strength of, measured by the double ring method, is given as o~ = K I F J d


where K~ is a geometrical factor, F h is bending force at fracture, and d is glass plate thickness. The fracture of massive bulk glass at a temperature of

Stress Reloxation Hodulus G(t,T,c) O ( t ) : ~G(s) T ds


(~(~1= 11~ Gls) ds o

g(®) : q = JG(s) ds "~ o


I [\,,,,,t,




Fig. 4. Stress r e l a x a t i o n m o d u l u s G as a function of time a n d its v a r i a t i o n with temp e r a t u r e (shift factor) a c c o r d i n g to the Williams L a n d e l - F e r r y equation. Shift factor: log a T = - c I ( T T g ) / ( T - T~ + c2) ( W L F - equation).


R. Br~ckner / Glass melts ut.Jer extreme forming processes

Fig. 5. High-temperature brittle fracture of a glass plate, 2 mm thick, 75 K above Tg; deformation speed: 49 mm s l; strength: 632 MPa (concentric double ring method).

about 75 K above Tg is shown in fig. 5 at a not particularly high deformation rate (49 m m / s ) [21]. The fractured surface looks really like a fracture at room temperature. At a lower deformation speed a viscous deformation would result instead of fracture. Therefore it is very important for the measurements of high-temperature strength, that the deformation speed is high enough, so that a viscoelastic deformation takes place not larger than that which is required for the pure elastic deformation within the linear theory. This is the case for the concentric double ring method, if, for the dimensions used here, the maximum deformation in the centre of the glass plate w0 is smaller than the thickness of the plate: w0 < d. Under this supposition one can get a strict separation between the brittle-elastic and the plastic-viscous behaviour of the glass melt. The separating curve corresponds with that minimum deformation rate, at which a brittle bending fracture takes place [21]. With increasing temperature this minimum deformation rate increases (fig. 6). The connection with viscosity is nearly linear, because the slope of the straight line in fig. 7 is 1.1. The values in fig. 6 correspond to three different "apparent Young's moduli" E*. For the minimum deformation rate the interval of E* values is 18 < E* < 25. In this case the stress-strain curve is not linear and E* is calculated as a mean value according to fig. 8 and equation

E* =K:F/(d3wo)


w0 < d,

R. Br~ckner / Glass melts under extreme forming processes












~ O

10 L ~-




100. v 10-1-







o% / 0

v 45 ~_ E"







18, E%25

ploshc- wscous





/ /






Iemper0turein °C Fig. 6. M i n i m u m deformation rate for brittle bending fracture of float glass with untreated surfaces at small bendings wo < d as a function of temperature. E* is the apparent Young's m o d u l u s (see text and fig. 8).

where F is fracture force, K 2 is constant depending on geometry and Poisson ratio, d is plate thickness, w0 is m a x i m u m bending of the plate. At higher d e f o r m a t i o n rates the curvature of fig. 8 will be less and therefore E* will b e larger. For these values, 25 < E* < 45 and E* > 45, the deformation rates for

'°'1 uo



brittle- elestic




g g

10 0 -





10-I_ 10.2.

/ o



1'011 Sheorviscositv




in dPos

Fig. 7. M i n i m u m deformation rate for brittle bending fracture of float glass as a function of viscosity.


R. Br~ckner / Glass melts under extreme forming processes






~, 60

% ~' 40








Bending in rnm

Fig. 8. Apparent Young's modulus E*, calculated from fracture stress and plate bending at fracture; example shown: deviation from linearity for float glass with grinded surface at 562°C; piston speed: 5 ~ m / s ; t~ at fracture = 0.25 M P a / s ; wo = 0.48 d.

fracture are above the critical curve of fig. 6, that means within the brittle-elastic region. Under this aspect the bending strength of float glass plates has been measured. The first results are shown in fig. 9 [21]. There is a very steep increase in strength ÷ around Tg depending on the deformation rate or on the apparent Young's modulus E*. The decrease in strength at about 60 K above



I 400-


45; E" D 25, E',45



o 18, E';25


d' Da








56o ~ ~6O in °C

760 "-

Fig. 9. Bending strength of float glass (F3), with untreated surfaces as a function of temperature and loading kinetics. Full circles: 8 = 2 M P a / s ; open symbols: higher rates. + Preliminary results of Shinkai et al. [22] show a similar tendency of fracture toughness: an increase below Tg for silica glass and an indication of that at Tg for float glass.

R. Br#ckner / Glass melts under extreme forming processes


Flootglos F4

400 E



(~ : 2~a/s


E" •


95 % confidence range of mean volue






250" 200

© ®










Temperature in °C 600I

I 500-

DurQn untreated •

6 = 2MPo/s



-" 10



95 % confidence range of mean value

~' 300g c~









560 ig in





Fig. 10. Bending strength of float glass with untreated surfaces as function of temperature showing no maximum in contrast to fig. 9. a) float glass: b) borosilicate glass.

Tg is possibly due to phase separation effects with or without crystallization. This suggestion is confirmed by fig. 10 (a,b) in which the results on float glass and D U R A N glass are presented [23]. These samples, which do not show a maximum, have not been additionally annealed in contrast to the samples of fig. 9. They show an increase in strength above Tg of > 100% for the float glass and of - 500% for the D U R A N glass. Even samples which have been grinded


R. Brf~ekner / Glass melts under extreme forming processes

I 250-

FloatglosF4 • ~, : zMPozs ,


E° -








95'/. ~ootid~,~ ~o,,ge



,.0, , r . . , . ,


/'~o , ~ /







1 •




50 - _ , . _ _ _ _ . - - - ~ ,



. . . .


"-~--" "--I'"'2"//1



Temperature in °C

5(}0 i~ 6(}o

7oo =-

Fig. 11. Bending strength of float glass as a function of temperature with grinded (triangles) and post-grinded heat-polished (at 600°C) surfaces (squares and diamonds).

at the surface show an increase in strength above Tg (fig. 11) but on a lower level [23]. Heat-polishing at a temperature of 600°C leads to a level between. The interpretation of these results may be gathered in two points. First, a healing process takes place during the period when thermal equilibrium is

Fig. 12. High temperature brittle fracture of a glass fiber in the viscoelastic range of the fiber drawing process.

R. Br~ckner / Glass melts under extreme forming processes


established at the temperature of strength measurement. This may be connected with a flattening of surface cracks (blunting of the crack tip and reducing the crack depth). Second, flow processes are involved, especially at the crack tips by which local stresses will be reduced during fracture. In that connection one has to take account not only of Newtonian, but also of non-Newtonian flow behaviour (see next section), for which a drastic decrease of viscosity is expected not just due to dissipative energy but mainly due to structural effects. Another example for brittle high-temperature fracture in the viscoelastic range of glass may be taken from the glass fiber drawing process. Figure 12 shows the brittle fracture of a glass fiber at a temperature above 1000°C [24]. The fiber drawing process was increased up to the limit of the strength of the fiber. Typically for that kind of fiber fracture is the flat geometry of the fracture surface in contrast to another kind of high-temperature fracture. Rekhson [25] has observed no less than three types of high temperature fracture of glass rods including the described type as well.

2.3. Non-Newtonian flow behaviour There are various exceptions from "normal" Newtonian flow behaviour and the reasons for that are quite different and very specific for the regarded glass forming systems. Therefore we may classify those systems by three principles: Non-Newtonian flow in stable melts above the liquidus temperature, metastable melts below liquidus temperature and unstable melts which show phase separation or, partly, crystallization.

2.3.1. Stable melts above fiquidus temperature Examples of this category are very rare. The reason for that may be that viscosity is relatively low. Therefore it is very difficult to introduce extreme flow conditions into the melts without heating them up by dissipative energy. One example may be given briefly from an extreme fiber drawing condition which gives one of the limits [24,26,27] of the fiber drawing process. Figures 13 and 14 show interruptions of fiber spinning at a position of the gob where the fiber has reached a relatively low temperature (fig. 15) and a high drawing stress (about 130 MPa or more) [24]. The fiber of originally 10 /~m diam. is drawn out to a diameter of about 100 nm by a reheating dissipative process, until a temperature is reached at which surface tension force exceeds viscosity force and interrupts the drawing process [26,27]. If one excludes dissipative temperature effects, and looks for non-Newtonian flow behaviour under less extreme conditions as described, he will find that only for those melts which consist of chain- or disk-like flow units. Not much has yet been done in this field. Therefore only one example is given. It is taken from the chalcogenide system sulphur-arsenic (fig. 16) [28]. In this system chain- a n d / o r disk-like flow units are the cause of structural viscosity behaviour. They are oriented by the action of shear stress and disoriented by

R. Br~ckner / Glass melts under extreme forming processes





10pro i

b) I 30Ohmj



10pro r


b) L_---.-.--J 30Ohm ~

Fig. 13. Ends of "fractured" fibers in two enlargements, drawn at 1293 K and 100 mbar nozzle pressure. Starting diameter of the fibers: 10/Lm.

the Brownian movement. As a result of that these melts show also a strong effect for flow birefringence (see sect. 3.3).

2.3.2. Metastable melts below liquidus temperature It may be of some interest to start with a more or less exotic effect of non-Newtonian flow. At high viscosities, when structural equilibrium is not yet






Fig. 14. Ends of a "fractured" fiber, which was bent after failure. Same conditions as in fig. 13.

R. Br~ckner / Glass melts under extreme forming processes

,150o .





1400- o-o~










/' ~

N 1200-



oo_./ 1100 .....




"~/ ' /T0=1293K/p--10mb°r

/ .-





II . . . .

0,5 0-



5'0 1[)0 Fiber tenqih L in pm

150 "--

Fig. 15. Calculated temperature distribution during the necking process and failure of E - g l a s s fibers, which have been d r a w n at 1 2 9 3 K nozzle temperature and 100 or 10 mbar nozzle pressure, respectively.




+ ~ 162°C


~1 ~a






197°C A 213°C ~D ~ O 234°C x 254oc

[3 x








Rotation speed (rev rain ~)

Fig. 16. Viscosity of AS2Sl00 melts as a function of rotation of the inner cylinder of a rotation viscosimeter.


R. Bri~ckner / Glass melts under e x t r e m e f o r m i n g processes

50 "70~ 3o'





' -

"~7o /

5~5°C 7,0

7,7 7,~





7,¢ 7,7 7,8 7,g g,#


Fig. 17. Viscosity as a function of strain or time, respectively, after ref. [2]. Samples with various heat pretreatments: l, after producing of the rodsample without pretreatment; If, after heat pretreatment at 575°C for 237 h; and with various stresses within the strain-range l / l o = 1.3-1.5 tenfold stress was applied.

been reached, viscosity depends not only on time as was shown in sect. 2.1, fig. 2, but also on stress or strain rate, respectively [2]. Stress-independent viscosities are obtained for equilibrium viscosities. However, before structural equilibrium is reached the viscosity depends on the applied stress (fig. 17). The higher the stress, the higher the viscosity if the starting viscosity is lower than that at equilibrium. That means a temporary dilatant viscous flow behaviour is the result. On the other hand, it would be interesting whether a structural viscous flow behaviour will be obtained during the time when the equilibrium viscosity will be reached from higher viscosities (samples with a fictive temperature below that of viscosity measurement). No measurements of this kind have yet been made. At very high stress levels, above about 130 MPa, viscosity depends on stress at all [29], on strain and strain rate [30,31]. Under tension as well as under compression a decrease of viscosity is obtained, that means a so-called structural viscous flow behaviour. Figure 18 shows some plots of viscosities normalized by the Newtonian viscosity (~/0) versus normalized strain rates by ~% (~ is strain rate, z0 is average shear relaxation time % = ~o/G, G is instantaneous shear modulus) [30]. Plot (a) is constructed with results from a rubidium silicate [29], plot (b) from a soda-lime-silica glass and plot (c) from a hypothetical molecular dynamics calculation on a Lennard-Jones glass. The measurements have been made under tensile stress of thin rods (0.1-1.0 mm) in a range of the Newtonian viscosity between about 10112 and 10 ~4 dPa s (Poise). All plots in fig. 18 can be described by the equation [30]: = (1 +

where a is found to be the ratio of the maximum stress sustained by the glasses and the instantaneous shear modulus o'limit//G. Olimit represents the actual cohesive strength of the glasses and is interpreted as the maximum stress sustainable by the material, or when a steady-state flow cannot just be

R. Br~ckner / Glass melts under extreme forming processes



(a) 1.0 ,,~\,=


°~\.~ G/o'limitl = 1 9 0 - ~


- O.C








Fig. 18. Plots of viscosity reduced by the Newtonian value versus normal-


uJ .J ¢ o z


G/(T(llmtt) : 1 6 0 ~ ° 0.( 1.0


--. o



\o %,°

GA:r(ImR) =13 - j

~\,~ o r~



0.OO 1




S T R A I N R A T E , I~ To


ized strain rates,, for various temperatures after ref. [30]; samples: thin rods or fibers under tensile stress, a) Rubidium-silicate data from ref. [29] at various temperatures: open circles - 528°C, solid triangles = 555°C, open triangle = 536°C, squares 501°C, solid circles = 480°C. b) Soda-lime-silica glass data from ref. [30]; circles = 563°C, diamonds = 574°C, triangles=593°C, squares= 596°C. c) Molecular dynamics calculations on a Lennard-Jones glass with different densities representing different thermodynamic states.

sustained. In the case of l o a d e d fibers the d i a m e t e r undergoes localized n e c k i n g until a p o i n t is reached when O~pp~ea > O'limit a n d fracture occurs, o,,,,, was f o u n d to go up to 4 - 5 M P a for a s o d a - l i m e glass [32]. N o n - N e w t o n i a n viscous flow is o b t a i n e d , too, when bulk glass m a t e r i a l (cylinders with 10 m m diam. a n d 12 m m height) are d e f o r m e d not u n d e r tensile, b u t u n d e r compressive stress [31]. F i g u r e 19 shows a drastic decrease of viscosity with increasing strain rate at t e m p e r a t u r e s a b o v e Tg. T h e original curves, f r o m which fig. 19 was o b t a i n e d , are p r e s e n t e d in fig. 20. A f t e r a steep increase of the a p p l i e d force a m a x i m u m , after that a m i n i m u m is o b t a i n e d . W i t h increasing speed of the p i s t o n first cracks occur; however, n o t at the d e f o r m a t i o n of m a x i m u m force, b u t at m u c h greater d e f o r m a t i o n s , after the m i n i m u m of a p p l i e d force. W i t h respect to the high m e c h a n i c a l energies a n d to the large samples, c o m p a r e d with the fibers above, one has to regard a t e m p e r a t u r e increase a n d therefore a viscosity decrease. In fig. 19 the viscosity values are taken from p a r t s of the s t r e s s - s t r a i n curves which are at the e n d of the d e f o r m a t i o n processes or at the occurrence of the first cracks, respectively. F o r these c o n d i t i o n s the viscosities agree relatively well with those c a l c u l a t e d from the m a x i m a l t e m p e r a t u r e increase due to the


R. Br~ckner / Glass melts under extreme forming processes


g~ • 581 =c v 596%




611 =c



63& *C 6/~oC


o ca

• =1 °

o 657°C


E z


t 10"

Compressive Strain Rctfe in s-1 Fig. 19. Logarithm of normalized viscosities of float glass at the end of the deformation processes or at the instant of the first cracks, respectively, at various temperatures as a function of axial strain rate.

! so /-......

,=o5,0 10g ~ = 8.33[



.~ !

o,~ aos


_..._.002 ~ ' ~ ~....~ 0,01~




Axiole compressive strotn In ( h o l h ) - - - ~ - , , - Fig. 20. Axial compressive stress as a function of axial compressive strain at various deformation rates. The points and dotted lines indicate flow at or after the first crack.

adiabatic heating of the whole specimen by complete conversion of deformation energy. In contrast to that a significant reduction of viscosity is observed at earlier stages of the deformation process when the work of deformation is still small and the possible increase in temperature is negligible. Thus, the diminution in viscosity must at least be due to non-Newtonian flow behaviour resulting from structural changes in the glass melt at high shear rates and stresses. This non-Newtonian flow behaviour (reduction of viscosity at high shear rates and stresses) may be considered as an advantageous fact which

R. Brf~ckner / Glass melts under extreme forming processes


reduces stress concentrations at crack tips and so reduces high-temperature fracture sensitivity of glass melts. 2.3.3. Unstable melts

Concerning the flow behaviour of phase-separated melts two main influences have to be regarded: - first, irreversible time-dependent viscosities; second, non-Newtonian viscosity within short time intervals (quasi-time-independent, but strain rate-dependent viscosity). Usually both effects are acting together in unstable systems as is the case in phase-separated amorphous a n d / o r partly crystalline glass melts. First case: From a general viewpoint an increase in viscosity with time will be expected, if the separating minor phase has the lower viscosity; a decrease is expected, if the separating minor phase has the larger viscosity. Usually this should be the case in all systems depending on which part of the mixing gap the melt composition is situated, on the left or on the right of the maximum decomposition temperature. Examples for increasing viscosities are more frequent in the literature [33-38], whereas examples for decreasing viscosities are very rare [36,37]. It seems, that the region of the latter case is only a very small one, ranging from the binodal of the phase with the smaller viscosity to the spinodal. The part of composition within the whole spinodal obviously seems to give that phase with the larger viscosity the dominating effect for the mean viscosity of the phase-separated system by aggregation. S e c o n d case: A very broad field of non-Newtonian flow behaviour should be expected in phase-separated glasses, which may be comparable to that of emulsions (amorphous separation) or of suspensions (crystalline separation). Thus, nearly all kinds of non-Newtonian flow behaviour can be expected, depending on the kind of phase separation (binodal or spinodal), on the kind of crystals, on concentration, particle size distribution and habitus of the crystals. However, astonishingly nothing is found in literature. Araujo et al. [39] and Seward [40] have certainly not measured the flow behaviour of phase-separated photochrome and other glasses, but they have used flow behaviour for producing textural effects in order to get orientation-dependent properties by freezing-in a certain flow condition. In this way they succeeded in getting photochromic polarizing glasses, that means birefringence and an orientation-dependent electric loss factor.

3. Anisotropy and orientation phenomena

In 1977 Takamori and Tomozawa [41] have given an excellent review on "Anomalous Birefringence in Oxide Glasses". To this article, its references, and additional articles [42-44] will be referred for more details and only additional references will be cited here. Takamori and Tomozawa summarized the results on birefringence in glasses, produced by viscous flow, in table 1.


R. Br~ckner / Glass melts under extreme forming processes

3.1. Birefringence in phase-separated glasses As in the case of phase-separated photochrome glasses (sect. 2.3.3) any phase-separated glass will cause birefringence if a deformation and orientation has been produced by uniaxial flow of the particles. This kind of birefringence is called form birefringence. Usually the effect is relatively high and is of the order of magnitude of several thousand n m / c m . The distribution birefringence is less and is the result of a different concentration of particles in two directions of the matrix. This kind of asymmetry can be obtained with anisometric as well as with isometric particles or droplets. One example for the latter case is the birefringence obtained after heat treatment of prepulled liquid-liquid phase-separated glasses. There is a great field of preparation and application of orientation phenomena in phase-separated glasses, which is extensively treated in refs. [39-43]. However, as can be seen from table 1, there is a lack of any information about frozen-in strain birefringence for phase-separated glasses and about the differential contraction of anisotropic phases. This may be a consequence of only few informations about frozen-in strain birefringence, orientation birefringence produced by anisometric flow units and contraction after annealing in singlephase glasses.

Table 1 S u m m a r y of anomalous birefringence in glasses [41] Classification

Structural requirement

Frozen-in strain Frozen-in strain (type II) Differential contraction of anisotropic phases Form birefringence


Chain orientation Anisotropic array of micropores

Direction of inducing stress

Sign of birefringence

Typical glass reported

Tensile Compressive

+ a ? ?

Soda-lime Flint None b None b

Phase separation Phase separation

Phase separation

Phase separation

Molecular structure Microporous

Tensile (elongation) Compressive Tensile (elongation) Compressive Tensile (elongation)

Borosilicate + Borosilicate + NaPO 3 + - c

Vycor-type microporous glass

a Most of the technical glasses, except dense flint type, have positive stress-optical coefficients. b These birefringences are quite feasible when a phase-separated glass is processed thermomechanically. No definite observation has been reported yet, however. c Along the direction of leaching.

R. Br~ckner / Glass melts under extreme forming; processes


3.2. Birefringence in single-phase glass fibers' The exceptional properties of glass fibers as compared with bulk glass, especially tensile strength and optical anisotropy, bring up the question of the difference between the structure of the fibers and that of the bulk glass. These two materials with the same chemical composition are characterized by a totally different thermal and mechanical history, which goes back into the molten state several hundred °K above the glass transition temperature. If all drawing conditions of the fiber production process are known [45,46], it is possible to produce different properties not only in fibers with the same chemical composition but also with the same fiber radius. Thus, there are differences among similar looking fibers, if the thermal and mechanical history is different. Much greater are those differences between fibers and bulk glass of the same chemical composition. The general relation between drawing stress and resulting optical birefringence is given in fig. 21(a) [47]. This master curve is valid for a constant nozzle temperature. Figure 21(b) illustrates the birefringence vs. stress behaviour at various nozzle temperatures. It is remarkable that the birefringence increases with nozzle temperature at constant stress levels. This behaviour indicates that the deformation and the orientation of the network fragments is easier at higher than at lower temperatures; and that this condition can be frozen in, resulting in an anisotropic structure. The density of the fibers indicates also a frozen-in structure. Figure 22 shows the fiber density as a function of the viscosity of the glass melt in the nozzle at constant mass flow. One would expect, that the fiber density decreases with decreasing viscosity, i.e. with increasing fictive temperature. However, fig. 22 illustrates the opposite, the density of the fibers increases with increasing nozzle temperature for fibers with a constant diameter, that means with constant cooling rate. This anomalous behaviour is due to the drawing stress, because at constant cooling rate and constant mass flow the stress is only a function of the temperature, high at low temperature, low at high temperature. Therefore the fiber density varies antipathetically with nozzle temperature. As a result of measurements of this kind, to which also belongs the shrinkage behaviour of the fibers at Tg (fig. 23), at least two components of the structure of glass fibers are evident [47]. The first component, inherited from the bulk glass, is an isotropic open structure, which becomes more open at higher temperature. It is modified by a second component, due to the action of the drawing force during the production of the fiber. This second component is anisotropic and is responsible for the anomalous properties of the fiber. The resulting structure of pure silica glass fibers, for example, is schematically shown in fig. 24, a more open structure in axial direction as compared with that in radial direction [48]. In contrast to silicate glass fibers with a three-dimensional network structure, which show a very small anisometric effect but a considerable anisotropic effect, are the phosphate glass fibers within a composition range, in which the

R. BriJckner / Glass melts under extreme forming processes

440 GO0



To 1473K





<~ 300 .E

200, 100. ,C~



rht C~nst


r~t COrrM v t t;otlst / R~


- -



c3nst /R,


t consl





Orowlng Stress o m MPo

120 ,-~

Fig. 21a. B i r e f r i n g e n c e o f E-glass fibers a t c o n s t a n t nozzle t e m p e r a t u r e (1473 K ) as a f u n c t i o n o f the d r a w i n g stress; p a r a m e t e r s : m a s s f l o w th, d r a w i n g s p e e d vz, f i b e r d i a m e t e r 2 R ; zx: rh = c o n s t a n t , vz ~ c o n s t a n t , 2 R E ~: c o n s t a n t ; © : rh ~ c o n s t a n t , vz = c o n s t a n t , 2 R E #: c o n s t a n t ; ) : rh c o n t s t a n t , v, q: c o n s t a n t , 2 R E = c o n s t a n t = 7 . 5 / t m .

chain structure dominates. Thus it is expected, that the anisometric effect will be very large and will produce high birefringence values by a strong orientation effect. As is shown in fig. 25 [49] the birefringence of lithiumsodium-metaphosphate glass fibers is by a factor of nearly 100 larger than that of the E-glass fibers at comparable drawing conditions [50]. These are values of

I 800 7~, S 600 e_ =


--o-- 1623 - - , - - 1573 --'~-- 1523 --a-- 1473 - - - - - 14Z3 - - - - - 1373




o./Z//,., 2




100 20

40 60 80 Orawing Stress or tn MPo

100 =

Fig. 21. (b) B i r e f r i n g e n c e o f E-glass fibers as a f u n c t i o n o f the d r a w i n g stress f o r v a r i o u s nozzle t e m p e r a t u r e s .

R. Bri~ckner / Glass melts under extreme forming processes


~° ?.5mgls



2.5Z8 -~



ZRt'n Pm 30~ -- 0.5 - - ~ - - 16.0 -- 0,5 -o--



--,-- 15g *_0.3

2.5184~ 2.0 - -

, 2.5 log

, ' , 3.O 3,5 ~0,11D m dPos

Fig. 22. Density of E-glass fibers as a function of viscosity of the glass melt in the nozzle at constant mass flow r~; parameter: fiber diameter.


the order of magnitude of those for organic polymers. There is a remarkable difference between the E-glass fibers (fig 21) and the metaphosphate glass fibers (fig. 25). While the birefringence of the former shows no dependence on mass flow and drawing stress, the latter is depending on mass flow and

Bulk G~mssI 1273 1073



g~ ~0 -1-





Heahng T~mem h ~


-4 2RE 5,0-. 03 pm

Fig. 23. Shrinkage of E-glass fibers as a function of heating time. Parameter: fiber diameter. The dotted line indicates the heating program.


R. Brftckner / Glass melts under extreme forming processes Direction of stress

Fig. 24. Schematic representation of a vitreous silica fiber structure.

drawing stress in a somewhat complicated way. At the highest nozzle tempera ture a deviation from linearity with increasing mass flow and stress to uppe~ values of birefringence is observed (fig. 25(a)). At the lowest nozzle temperature a deviation to lower values is obtained (figure 25(b)). This behaviour has to do with the formation process of the phosphate chains in that temperature range as will be apparent from flow birefringence measurements (see the next section).

3. 3. Flow birefringence in glass melts When a liquid phase is strained under the action of a stress field it shows birefringence, if the molecules or the structural units of that liquid phase are not isometric in their habit. Because these units, also called flow units, will be oriented by the mechanical stress field, as produced for instance by a shear velocity gradient, as a result the liquid shows birefringence. Already, Maxwell [51] has defined the specific birefringence D in which the maximum birefringence A n max is related to the refractive index n of the liquid and to the shear stress, which is the product of the viscosity 71 and the shear velocity gradient u. Theoretically Raman and Krishnan [52] calculated the specific birefringence D from structural units, in which M is the molecular weight, L the Loschmidt number, r the density, n the refractive index and f a form function of the more or less anisometric molecules or flow units. The relatively complicated function f can be simplified as indicated below [53]. A distinction can be made for three ranges of the form function: f is zero for a liquid with isometric flow units; for f smaller than 0.1 the flow units will deviate, but not too far from a spherical habit; for f from 0.1 to 0.5 the flow units will be disk-like and for f values between 0.5 to 2.0 the flow units will be chain-like.










----tensJJe slress m MPO




riO0 °C





i I

















• ~8"

50 tensde stress in M ~



P : 1 4 0 o ~ ~ 9 0 tabor

ZSNazOZSLI:O 5OP:Ot 375 °C

• :f


Fig. 25. (a) Birefringence of mixed alkali-metaphosphate glass fibers as a function of the tensile stress resulting from the drawing stress. Parameter: mass flow: nozzle temperature 500°C. (b) Birefringence of mixed alkali-metaphosphate glass fibers as a function of the tensile stress resulting from the drawing stress. Parameter: mass flow; nozzle temperature 375°C.








R. Bri~ckner / Glass melts under extreme forming processes



o Sulphur 0 As2S2~o A"As2S2oo

• 1 500"

• As2SIso

8 o

• As2Stoo As2Sso s


As2S6o s


As2S3o ~





180 200 220 240 Temperature CC)


2~0 36o

F i g . 2 6 . R e l a t i v e b i r e f r i n g e n c e o f v a r i o u s a r s e n i c s u l p h i d e m e l t s as a f u n c t i o n o f t e m p e r a t u r e .



A/'/max/(nT / grad u ) = [(n 2 - 1)(n 2 +


0 < f < 0.1 spheroidal flow units, 0.1 < f < 0.5 disk-like flow units, 0.5 < f < 2.0 chain-like flow units. For a large number of organic liquids it was shown that the values of the form function, calculated from the molecular structure and from the results of the streaming birefringence, agree very well, except those of the alcohols. The structure of the alcohols should give a theoretical form function for chain-like units, whereas the specific birefringence indicates an experimental form function for spheroidal units. The reason for this apparent discrepancy is that alcohol chains form agglomerates resulting in spheroidal flow units [53]. The streaming or orientation birefringence method has been applied to B203 and to a sodium-borate melt [54]. It was found, that no disk-like BO 3 groups exist as flow units, but, as in the case of the alcohols, spheroidal units were found up to a temperature of 1000°C. On the contrary, in sulphur and in arsenic sulphide melts chain- and disk-like flow units have been obtained from the specific flow birefringence depending distinctly on the temperature and on the arsenic concentration [55]. As in the case of the famous viscosity maximum of the sulphur melt the streaming birefringence shows a similar maximum as a function of temperature (fig. 26). This maximum first decreases, then increases with increasing arsenic content. Combined with the viscosity, density and refractive index as a function of temperature one obtains the Maxwell-constant D, which gives together with the form function f the possibility for estimating the mean molecular weight of the melts at various temperatures. This is demonstrated in fig. 27 [55] and is roughly valid for the sulphur melt, because M~ regards only the portion of the chains, according to the fact, that the spheroidal Ss-rings of

R. Br~ckner / Glass melts under extreme forming processes



C Sulphur

• As2S,o o ~7 As~Sm~ • As2S~o x As2S~






~7/v'V -



10 2 1"8

V~ V


~ x

I. 20

I 2'2

I 2-4

1 2-6

Reciprocal of absolute temperature x 109

Fig. 27. Mean molecular weight, M~, of sulphur and arsenic sulphide melts as a function of the reciprocal temperature with a form function of 0.5 calculated using viscosity values extrapolated for very high velocitygradients. The values for the As283o.5 melt where also calculated for f = 0.1. sulphur do not give a contribution to birefringence. For the other curves of fig. 27, it seems that no systematic trend exists. However, one has to regard that not only M r, but also f is changing with arsenic concentration. Thus, the up and down of the curves for the arsenic sulphide melts is due to changes of f from 0.5 to smaller values, that means changes of structure from long chains over small chains and two-dimensional disk-like to three-dimensional flow units. Similar effects have been obtained recently from metaphosphate melts [56]. The streaming birefringence indicates that after an as yet unknown intermediate structure-forming process the formation of chains starts at relative low temperatures (350-400°C) and high viscosities (103-104 dPa s), respectively. The chain-forming process increases rapidly as is shown with the product M . f in fig. 28. This process continues up to large viscosities and the viscosity itself (fig. 29) as well as the birefringence A n become strain- and time-dependent (fig. 30).

4. Outlook for the year 2004 From the preceding treatment and from common knowledge of these topics, it follows that many questions are still open and have been opened to new questions for which answers may or may not be found. Without any demand upon completeness some problems may be mentioned, which might be treated or apparently be solved within the next decennia:

R. BrVtckner / Glass melts under extreme forming processes


1500o E

,, Gloss T o Gloss II


LiNo0/Pz05 = 1,13 LiNa0/Pz05 : 0,9g

E o




500" cJ



g co




360 lemperoture in °C

Fig. 28. Product of mean molecular weight and form function f of mixed alkali-metaphosphate melts as a function of temperature.

11,72 -, 11,70 ~, T = 2/*2,5 °C


Gloss I

LiNe0 / Pz 0s = 1,13


11,66 x

0 I= -


co {.3 t~ >

11,64 11,62 11,60 0

11,58 11,56 11,54

16 - -

z'0 Elongohon




in %


Fig. 29. Log 7/ viscosity of metaphosphate glass rods as a function of strain at constant tensile stress.

R. Br~ckner / Glass melts under extreme forming processes


f = 245°C







14- 6~6 ~ ~


Grass I


LiNe0/P z05 = 1,13

2 !




513 Elongation At/l



in %

Fig. 30. Birefringenceof metaphosphateglass rods reduced by the applied stress as a function of strain at constant stress.

a) There is still a lack for a complete exact theory of viscosity. The various concepts are still semi-empirical, even the free volume concept, which gives a plausible explanation for the Williams-Landel-Ferry equation and for the V o g e l - F u l c h e r - T a m m a n n equation. Both equations are mathematically identical, the latter found in a purely empirical manner already in 1921. However, despite of this fact and lack, many qualitative structural conclusions could be drawn, but very seldom in a quantitative way. b) One distinct problem is the time dependence of viscosity on stress. Especially, the following question is not yet answered experimentally. From fig. 17 follows a dilatant or rheopexial flow behaviour during the time before the metastable equilibrium viscosity is obtained. The question arises for samples with a fictive temperature lower than that of viscosity measurement whether a dilatant/rheopexian or rather structural viscous/thixotropic flow behaviour will be obtained. c) Although two mathematical model descriptions have been applied to the non-Newtonian flow at high stresses or high strain rates, respectively, a physical and structural description or a physical interpretation of the applied constants is still open. Nevertheless, the two approaches from the Eyring theory [29] and from the Lennard-Jones potential [30] are very promising for further developments. d) Concerning the increase in strength at and above T~ the questions arise: how far can this increase be driven? At which level does the increase turn to the expected decreasing strength of the viscous melt as a function of temperature [57,58]? Which theories may be applicable? e) Birefringence measurements are promising in connection with flow, since


R. Br~ckner / Glass melts under extreme forming processes

" b i r e f r i n g e n c e tells us things a b o u t structure at all levels from the atoms up to those that scatter light, too, severely to m a i n t a i n coherence of the p r o b e b e a m " (Ernsberger cited in ref. [41]). So far, only a limited n u m b e r of glass scientists have investigated in this area a n d " m a n y questions r e m a i n to be answered in future work" [41]. This is in particular valid also for birefringence a n d structure of glass fibers. f) Especially, flow birefringence is a powerful tool in a d d i t i o n to viscosity for evaluating structural p r o b l e m s of glass melts. Concrete special work should be d o n e in this area, too. Besides flow birefringence also its relaxation time will be of great interest. As c o m p a r e d with viscosity, flow birefringence brings more light into the structure of glass melts t h a n viscosity. g) M a n y detailed p r o b l e m s m a y be listed. F r o m a general viewpoint more intensive a n d systematic research should be done in the field of fluids a n d melts; not only h y d r o d y n a m i c s , b u t also structure a n d properties should be investigated. Similar to the field of "Physics a n d Chemistry of the Solid State" the scientific field of "Physics a n d C h e m i s t r y of the Liquid State" should be founded. M a y b e this research will be pushed forward stronger d u r i n g the next 20 years.

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

H.R. Lillie, J. Amer. Ceram. Soc. 16 (1933) 619. H.J. Oel, Glastechn. Bet. 33 (1960) 219. K. Endeil and H. Hellb~gge, Angew. Chem. 53 (1940) 271. J. O'M. Bockris and J.D. Kitchener, Trans. Faraday Soc. 51 (1955) 1734. C.R. Kurkjian and R.W. Douglas, Phys. Chem. Glasses 1 (1960) 19. L. Shartsis, W. Capps and S. Spinner, J. Amer. Ceram. Soc. 36 (1953) 319. [7] P.J. Bray and J.A. O'Keefe, Phys. Chem. Glasses 4 (1963) 37. 18] P.J. Bray, M. Leventhal and H.O. Hooper, Phys. Chem. Glasses 4 (1963) 47. [9] G.E. Jellison and P.J. Bray, J. Non-Crystalline Solids 29 (1978) 187. [10] R. Br~ckner, H.-U. Chun and H. Goretzki, Glastechn. Ber. 51 (1978) 1. [11] R. B~ckner, H.-U. Chun and H. Goretzki, Proc. Int. Conf. on X-ray and XUV spectroscopy, Sendai, Japan, 1978, J. Appl. Phys. 17 (1978) 291. [12] Y. Bottinga, D.F. Weill and P. Richet, in: Thermodynamics of Minerals Melts, eds., R.C. Newton, A. Navrotsky and B. Wood (Springer, Berlin, Heidelberg, New York, 1982) p. 207. [13] K. Hunold and R. Bri~ckner,Glastechn. Ber. 53 (1980) 149. [14] G. Urbain, Y. Bottinga and R. Richet, Geochim. Cosmochim. Acta, in press. [15] G. Gehlhoff and M. Thomas, Z. Techn. Phys. 6 (1925-26) 544. [16] I. Kushiro, J. Geophys. Res. 81 (1976) 6347. [17] W.A. Weyl and E.Ch. Marboe, The constitution of glasses. A dynamic interpretation, Vol. II, part I (Wiley, New York, London, Sydney, 1964) p. 672. [18] C.R. Kurkjian, Phys. Chem. Glasses 4 (1963) 128. [19] D.C. Larsen, J.J. Mills and J.L. Sievert, J. Non-Crystalline Solids 14 (1974) 269. [20] J.D. Ferry, Viscoelasticproperties of polymers, 2nd ed. (Wiley, New York, 1970). [21] P. Manns and R. Br~ckner, Glastechn. Bet. 56 (1983) 155. [22] N. Shinkai, R.C. Bradt and G.E. Rindone, J. Amer. Ceram. Soc. 64 (1981) 426. [23] P. Manns and R. Bri)ckner, to be published. [24] R. Br~ckner, G. P~ihlerand H. Stockhorst, Glastechn. Ber. 54 (1981) 65. [25] S.M. Rekhson, M.L. Grant and H.F. Peckman, Glastechn. Ber. 56K (1) (1983) 408.

R. Br~ckner / Glass melts under extreme forming processes

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[42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58]


M. Stehle and R. B~ckner, Glastechn. Ber. 53 (1980) 130. M. Stehle and R. Briackner, Glastechn. Ber. 54 (1981) 205. K.-D. Pohl and R. Briickner, Phys. Chem. Glasses 22 (1981) 150. J.H. Li and D.R. Uhlmann, J. Non-Crystalline Solids 3 (19970) 127. J.H. Simmons, R.K. Mohr and C.J. Montrose, J. Appl. Phys. 53 (1982) 4075. P. Manns and R. Bri~ckner, in: Proc. NATO-Workshop on Strength of Glass (1983) ed., C.R. Kurkjian, in press. J.H. Simmons, R.K. Mohr and C.J. Montrose, J. de Phys. Coll. C9, suppl, no. 12 (1982) 439. M. Watanabe and R. Koyama, J. Soc. Glass Technol. 41 (1958) 275, 316. J.H. Simmons and S.A. Mills, J. Amer. Ceram. Soc. 57 (1974) 109, J.H. Simmons, S.A. Mills and A. Napolitano, J. Non-Crystalline Solids 14 (1974) 302. P. Bernheim and A.C. Chaklader, J. Non-Crystalline Solids 5 (1970/71) 328. O.V. Mazurin, V.P. Kluyer and G.P. Roskova, Phys. Chem. Glasses 11 (1970) 192. M. Sardag and H.J. Stevens, in: Proc. 13th Int. Congress on Glass, Poster Abstracts (Deutsche Glastechn. GeseUschaft, Frankfurt/Main, 1983) p. 129. T. Takamori and M. Tomozawa, Phys. Chem. Glasses 21 (1980) 189. R.J. Araujo, W.H. Cramer and S.D. Stookey, US-patent 3 540 793. T.P. Seward 111, J. Non-Crystalline Solids 15 (1974) 487. T. Takamori and M. Tomozawa, in: Treatise on Materials Science and Technology. Vol. 12, Glass 1: Interaction with Electromagnetic Radiation (Academic Press, New York, London, 1977) p. 123. M. Tomozawa and T. Takamori, J. Amer. Ceram. Soc. 63 (1980) 276. M. Takata, M. Tomozawa and T. Takamori, J. Amer. Ceram. Soc. 64 (1981) 275. T. Takamori and M. Tomozawa, J. Amer. Ceram. Soc. 65 (1982) 9. M. Stehle and R. Bri~ckner, Glastechn. Ber. 50 (1977) 307. M. Stehle and R. Briickner, Glastechn. Ber. 52 (1979) 82, 105. H. Stockhorst and R. Briickner, J. Non-Crystalline Solids 49 (1982) 471. R. Bri~ckner, Comp. Rend., VII e Congr6s Intern. du Verre, Brussels (1965) 1.3.2, Nr. 38. H. Stockhorst and R. Brtickner, work in progress. H. Stockhorst and R. Briickner, J. de Phys. Coll. C 9, suppl, no. 12. Tome 43 (1982) 451. J.C. Maxwell, Proc. Roy. Soc. London A22 (1873) 46. C.V. Raman and K.S. Krishnan, Phil. Mag. 5 (1928) 769. H.H. Kas and R. Briickner, Rheol. Acta 5 (1966) 6. R. Bri~ckner and H.H. K~is, Glastechn. Ber. 38 (1965a) 473. K.-D. Pohl and R. Briackner, Phys. Chem. Glasses 23 (1982) 23. R. W~.sche and R. Briickner, work in progress. J.C. Fisher, J. Appl. Phys. 19 (1948) 1062. M. Kornfeld, Elastizit~t und Festigkeit der Fli~ssigkeiten VEB Verlag Technik, Berlin 1952).