Variation of Densities and Refractive Indices in Tempered Glass ROBERT GARDON*

Research Staff, Ford Motor Company, Dearborn, Michigan 4812 I

A knowledge of density distributions in tempered glass is helpful in better understanding the tempering process. Since the densi- ties and refractive indices of a glass are uniquely related, density distributions were determined by a schlieren technique which gives a direct measure of index gradients. The advantages of this nondestructive approach are that it yields the location of each density fraction within the glass and that it is equally applicable to fragments and whole pieces of tempered glass. Thus, it also allows observation of changes in local densities which accom-

pany fracture and the release of internal stresses.

11. Experimental Method

This requirement could be met by the schlieren method for measuring refractive index gradients, since the refractive indices and densities of a given glass are uniquely related." The method has the advantage that, since it measures refractive index gradients, it is particularly well suited for measuring relatively small differences of indices, or densities, which it would be harder to obtain accurately from absolute measurements. The measurements of interest involve index variations in one dimension only, so that one can advanta- geously use the schlieren system developed by Philpot4 and S w e n ~ s o n , ~ which photographically produces a curve of refractive index gradient (dnldx) vs position (x). Its quantitative interpretation is thus very much simpler than that of conventional, 2-dimensional schlieren observations. The Philpot-Swensson system is widely used in ultracentrifuges for observing the movement of sedimenta- tion boundaries; the optical system of such a centrifuge" was used in the present study.

~i~~~~ 1 illustrates the operation ,,fthis schlieren system, parallel light from a horizontal entrance slit passes through a specimen, with the slit at right angles to the refractive index gradient. A plate with an index gradient is optically equivalent to a prism. Light passing through various parts of the specimen is therefore deviated verti- cally by small angles propofiional to the optical path length through the specimen and to local refractive index gradients. The entrance slit is imaged on a diaphragm with an inclined slit. Thus, the only light that travels beyond this part of the optical system is that one ray from each flat bundle of rays which to the of intersection ofthe image of the entrance slit and the inclined slit in the diaphragm. Any veaical displacement of the image of the entrance slit by refractive index gradients in the specimen in a horizontal movement of this intersection.

On the screen. This image is diffused horizontally by the cylindrical lens, so that only vertical about the specimen is retained. The cylindrical lens is so positioned that it also images each point in the inclined slit as a vertical line occupying the appropriate horizontal position. The overall result is the projection onto the screen of a graph of vertical displacements of the image of the entrance slit, plotted horizontally, against vertical position in the specimen. To an appropriate scale, this is also a graph of local index gradients as a function of position. The required scale factor (the sensitivity of the

system) can be calculated from its geometry. The system can be calibrated experimentally by placing a prism having a small and accurately known wedge

tion. Once index gradients are known, the distribution of refractive indices in the specimen is obtained by in- tegration.

By varying the angle be- tween the inclined slit and the horizontal, the sensitiv- ity of the system can be varied. Thus it is equally

L useful for measuring the Entrance large angular deviations Sllt produced by a relatively

long light path through an

I. Introduction

VRIATIONS in the density of tempered glass have been noted and measured before. In a paper describing a centrifuge for

classifying crushed glass according to density, Plumat' showed the cumulative density distributions of two specimens of the same glass: one fine-annealed, the other severely quenched. The densities in the annealed specimen varied 0.0025 g/cm3, or 0.1%. This variation was presumably due to heterogeneity OfcomPOsition. The quenched specimen of the same glass was, on the average, 0.47% lighter than the annealed glass, reflecting an overall expansion of its structure; the spread of its densities was 0.22%, indicating a heterogeneity Of structure superimposed on the heterogeneity of

Such data prompted the question (cf. Acloque') as to what in- Of a heteroge-

neous density distribution in glass and the more-or-less simultane- ous establishment of temper stresses. To address this question experimentally, it seemed necessary, at the very least, to determine not only the spread of densities,' but also the location within the glass of each density fraction.

Measuring spatial density distributions presented a particular problem. In principle, one could measure the density (Or refractive index) of thin sections from various locations in the glass. In practice, this is laborious and not sufficiently precise. Another approach was clearly needed, preferably one applicable to tempered glass before as well as after fragmentation.

there may be between the

The camera lens produces an image of the

Received December 13, 1976; revised copy received May 30, 1977. *Member, the American Ceramic Society. *Spinco Div., Beckman Instruments, Inc., Fullerton, Calif.

L c Screen or Film angle in the specimen posi- OI

o w -4- .- E E

Fig. 1. Schematic arrangement of schlieren system.

143

144 Journal of The American Ceramic Society - G a r d o n Vol. 61, No.

Table I. Characterization of Tempered Specimens for Schlieren Study

Avg Temper produced* Heat

transfer Mid-plane Surface density,

No. fcallcm* s "C) inmlcm) (nrnlcm) h J h , fe/cm3) Specimen cceff., h tension, h.,, Compression, I?,, Ratio, pc*r

1 0.0047 1408 3112 2.21 2.5092 2 0.0089 1886 3998 2.12 2.5086 3 0.0029 1055 2363 2.24 2.5099 4 0.00023+ 363 773 2.13 2.5116 5 Annealedt 2.5147

'*Expressed in terms of stress-induced birefringence. ?Natural convection. $By cooling at 500"Cih.

unbroken specimen of tempered glass and the very much smaller deviations corresponding to the shorter light path and smaller index gradients in a fragment. Because of its high sensitivity, the method requires that the glass used be free of index gradients due to ream. Also, the entrance and exit faces of the specimens (i.e. the edge surfaces) must be plane and parallel. For this it is sufficient that the specimens have squared-off edges before tempering. The slight distortion of the edges on tempering does not measurably affect results. On the other hand, the edges of fragments must, of course, be reground and polished before measurements are made on them.

111. Results

After an initial exposure of regular plate glass to the schlieren test, all further work was done with specimens cut from plates of a well-homogenized optical glass," the relevant physical properties of which are close to those of plate or float glass. Strips of glass 10 cm long by 2.5 cm wide by 0.73 cm thick were quenched from an initial temperature of 677°C at various rates, with the results shown in Table I. The quench head consisted of nozzles 0.64 cm in diam. in a square array at 5 cm centers, The nozzle-to-glass spacing was 6.4 cm. The coolant was air, and heat transfer coefficients were controlled by varying the air pressure.'j Schlieren measurements were then made on specimens as tempered, and on suitably prepared fragments of these specimens.

Figure 2(A) is typical of the schlieren photograms obtained when looking edgewise through the 2.5 cm width of a tempered specimen. The two vertical lines are traces ofthe plate surfaces. The horizontal lines on either side represent zero index gradient; the vertical scale of index gradients is determined by the optical path length through the specimen and the setting of the schlieren system. The curves are plots of refractive index gradient as a function of position across the thickness of a tempered plate. Since tempered glass is the seat of birefringence as well as index gradients, two such curves were obtained in the schlieren photogram of Fig. 2(A). These curves apply, respectively, for light polarized parallel and perpendicularly to the plate surface. Interpretation of this photogram is illustrated in Figs. 2(B) to 2(E).

Figure 2(B) is a plot of the average index gradient, given by

dnldx = 1/s(2dnll/dx +dn .ldx) (1) This equation takes account of the fact that stresses in a tempered glass plate are 2-dimensionally isotropic. Thus dnll/dx applies for two axes in the plane of the glass, while dn ,/dx applies in the third dimension.

Figure 2(C), obtained by integration of Fig. 2(B), shows the variation of refractive indices across the specimen. In a separate experiment, similar to Ri t land '~ ,~ the refractive index vs density relation of the glass was found to be

dnldp=O. 176 cm3/g (2)

*No. 1045 Ophthalmic Crown, PPG Industries, Pittsburgh, Pa. Composition: 70.5 wt% S O 2 , 11.9 CaO, 8.6 Na,O, 7.7 K,O, 1.08 Sb,O-,, 0.2 SO:,, and 0.01 Fe,O,,. Room temperature values of Young's modulus 675,000 kglcm2, Poisson's ratio 0.22. and stress-optical coefficient 0.38 (kg/cm2)/(ndcm). In the transformation range (500 to 600°C): estimated values of thermal conductivity 0.0025 cal/cm s "C, and volumetric specific heat 0.8 cal/cm3 "C. Absorption coefficients 0.3 cm-' for I < h<2.7 y m , 5.0 crn-' for 2.7<h<4.5 y m , and 3: for h>4.5 y m .

I 4 Z

Schl ieren Photogram

0.004

0.00 2

0

,0004 a

Index 8. Density(relative1

Birefringence & Stress

I-- 0.73 cm-/ glass thickness

Fig. 2. tempered glass (specimen No. 1).

Interpretation of schlieren measurements on

Hence, to a different scale, shown on its right side, Fig. 2(C) also represents the variation of densities across the tempered glass specimen. At this stage absolute values of density (or index) are not yet known.

Having yielded separate index gradient curves for nil and nL , this schlieren technique also permits a determination of birefringence and hence stress. Thus Fig. 2(0) shows the birefringence gradient, obtained by the relation

dbldx =dn,,ldx -dnl ldx (3)

The variation of birefringence across the plate is obtained by inte- gration ofdbldx (solid line in Fig. 2(E)). Thedashed line represents conventional birefringence measurements, using a polarizing mi- croscope with a Babinet compensator. Although there is some difference between the gradients of these two curves near the sur- faces of the specimen, where the schlieren photograms are rather hard to interpret, there is very good agreement between the two regarding the magnitude of birefringence and, correspondingly, of frozen-in stresses in the glass.

When a tempered plate is broken, the fragments are irregular prisms, the end faces of which are parts of the original surfaces of the plate. It is possible, therefore, to cut small parallelepipeds from

Mar.-Apr. 1978 Variation of Densities and Refractive Indices in Tempered Glass I I I I

2.514-

-

145

I I I I - readily interpreted as a stress. An estimate of the

distribution such as b2'. What is clear is that p., the

3000 - residual stress distribution in the fragment might thus be represented by an equivalent birefringence

density distribution measured in the fragment, is

Fig. 3. Density and birefringence (stress) distributions (specimen No. 1).

selected fragments, which can then be examined in the schlieren apparatus in the same manner as the original specimen. The light paths through the fragments are much shorter, typically only 1 or 2 mm, depending on the degree of temper and the fineness of the resulting break pattern. To compensate for this reduction in path length, the sensitivity of the schlieren apparatus may be increased by decreasing the inclination of the slotted diaphragm relative to the entrance slit.

Interpretation of schlieren photograms of fragments parallels that illustrated in Fig. 2 for unbroken specimens, except that, with the substantial release of temper stresses on fracture, only a single curve ofdnldx is obtained. Thus, no birefringence is discernible from the schlieren data. However, direct measurements with a Babinet com- pensator do show some residual birefringence in the fragments, which can be quite pronounced, especially in larger fragments of less strongly tempered glass.

All results on specimen No. 1 are pulled together in Fig. 3. Figure 3(A) shows the density distributions as measured in the unbroken specimen (pl) and a fragment (pz), disposed about their common average density. Measurement of the latter, by Archimedes' method applied to the whole specimen, allowed the density scale to be made absolute. Figure 3(B) shows conventionally measured birefringence distributions in the whole specimen (b,) and the fragment (b2).

IV. Discussion

Perhaps the most striking aspect of Fig. 3(A) is the reversal of density gradients in temperedglass when it is broken. Thus, surface layers of the unbroken tempered specimen are more dense than the interior, but the opposite is true for fragments. The lower density of surface layers of the fragment is in accord with the view that more rapid quenching freezes in a more highly expanded structure, cor- responding to a higher fictive ternperat~re .~ In the unbroken glass there is superimposed on this true (i.e. stress-free) density distribu- tion a stress distribution (cf. curve b, in Fig. 3(B)) that causes dilatation of the interior of the glass and compression of its surface layers. This compression is evidently large enough to more than offset the locally lower stress-free density.

While the birefringence distribution bl in the unbroken specimen can readily be interpreted as a stress distribution, this is not the case for the residual birefringence distribution bz in the fragment. Any residual stress system in a small fragment would necessarily have to be treated as 3-dimensional; and bz certainly does not show the balance of tensile and compressive components expected with the essentially 2-dimensional stresses in a tempered plate. Only in the compression layer near the original surfaces of the plate, where stresses are most nearly 2-dimensional even in a fragment, can b2 be

of 677°C at dikerent rates (Table I). The leh half of Fig. 4 shows the density distributions p, as mea- sured in the unbroken specimens; the right, the stress-free density distributions p3, as calculated

from the initial densities for a total release of temper stresses. Stress-free densities in the rapidly quenched samples (Nos. I to 3) are lower in the surface layers than in the interior, and the average densities of all samples decrease with increasing cooling rate. Den- sity distributions in unbroken tempered glass ( p , ) are characteristi- cally different: here the surfaces are more dense than the interior. This is due to compression of surface layers and dilatation of the interior by temper stresses, which evidently more than offset the opposite distribution of stress-free densities.

'"For a large elastic plate under 2-dimensionally isotropic stresses, local changes in density are given by

~ ficrrD;) fip(x) = + 2(1-2v)

P E

where Sp=local change in density as a result of stress release; h = l o c a l changc in mess , with positive &T denoting an increase in compression or a rclease of tcnsion; E = Young's modulus; and u = Poisson's ratio.

(Annealed Glass)

average densities by thin horizontal lines Numbers identifv smcirnens

2.512 - n 5 \ 9,

Q 2.510 >;

u

c .- In c 0 n

2.508

2.506

0.5 O x/x, 0.5

I- 0.73 cm-4

Fig. 4. Density distributions in tempered glass.

146 Journal of The American Ceramic Society -Narayanaswamy Vol. 61, NO. 3-4 Finally, it is interesting to consider to what extent these experi-

mental observations support and, in turn, are explained by Narayanaswamy’s quantitative treatment of volume relaxation ef- fects in tempering.8 His calculations follow the thermal history of the glass as it is being cooled and use data on volume relaxation rates to determine the fictive temperatures at which different layers of glass set. His Fig. 8 illustrates how a distribution of fictive tempera- tures develops in tempered glass. From such differences between the fictive temperatures of different layers and the structural expan- sivity (Plzq-Pscass) of the gtass, density distributions were calcu- lated. These distributions are shown in the right half of Fig. 4 by the individual point symbols superimposed on the (continuous) plots of experimental data.* Agreement between the two is evidently very good.

V. Conclusion

A schlieren technique was used to measure refractive index and density distributions in specimens of tempered glass and its frag- ments. It was found that temper stresses are not wholly released even in small fragments, but density distributions in truly stress-free tempered glass could be obtained readily from the measured density and stress distributions.

The results present a clear picture of density distributions in tempered glass, both before and after fracture; and the difference between them is accounted for by the intervening release of stress. Thus surface layers of tempered glass are more dense (and have a higher refractive index) than the interior. This is due to compression of the surface layers and dilatation of the interior by temper stresses.

*The calculations of Ref. 8, like the schlieren experiments, yield only distributions of density within a given specimen, not absolute densities, which had to be measured separately.

However, density differences are smaller than might be inferred from the magnitude of temper stresses. The reason for this is that intrinsically, i.e. in its stress-free state, tempered glass has a non- uniform density distribution and its surface layers are less dense than its interior.

Experimental results on terminal density and stress distributions do not, by themselves, shed light on the tempering process. How- ever, questions regarding the interaction of density changes and the simultaneous genesis of temper stresses have meanwhile been an- swered theoreticallys; the present data on stress-free density distributions support and, in turn, are explained by this quantitative treatment of volume relaxation effects in tempering.

Acknowledgments: Acknowledgment is made to Paul Acloque, former Di- rector of Fundamental Research of Saint-Gobain, whose writings first aroused the writer’s interest in the question of possible structural effects in tempering. This is the question addressed experimentally in the pment paper and answered theoretically in the following paper by 0. S. NarayanasWamy, whose long-time collaboration on this topic is also acknowledged. The writer is indebted to Albert Naplitano, formerly of the National Bureau of Standards, forthe independent determination of thedensity vs index relation of ophthalmic glass and to Martin Goldstein of Yeshiva University for several helpful discussions.

References E. Plumat, “Study of the Homogeneity of Glass by Centrifugation in Dense

Paul Acloaue. “Review of Ptueress in Theories of Annealine and TemDering.” Liquids,” Verres Refract., 7 [ I ] 5-14 (1953).

L I

ibid., 5 [5 ] 243-60 (1951).

Constant Temperature,” J. Am. Ceram. Soc., 38 [2] 86-88 (1955).

Nature (London), 141, 283-84 (1938).

H. N. Ritland, “Relation Between Refractive Index and Density of a Glass at

J . St. L. Philpot, “Direct Photography of Ultracentrifuge Sedimentation Curves,”

H. Swensson, “Theory of Method of Measurement with Crossed Slits,” Kolloid Z., 90, 141-56 (1940).

Heat Transfer. ASME, New York, 1962.

Crown Glass,” ibid., 37 [8] 370-78 (1954).

this issue, pp. 146-52.

Robert Gardon and J . Cobonpue; pp. 454-60 in International Developments in

’ H. N. Ritland, “Density Phenomena in the Transformation Range of a Borosilicate

0. S . Narayanaswamy, “Stress and Structural Relaxation in Tempering Glass”;

Stress and Structural Relaxation in Tempering Glass 0. S. NARAYANASWAMY*

Research Staff, Ford Motor Company, Dearborn, Michigan 48121

Temper stresses are brought about, primarily, by a partial relaxation of transient stresses generated by rapid cooling of the glass. Stress relaxation under nonisothermal conditions is com- petently handled by a mathematical tempering model, in which glass is treated as a simple viscoelastic material. However, this model proved inadequate in some respects since the properties of glass depend not only on its instantaneous temperature but also on its prior thermal history. A tempering model was there- fore developed that incorporates both stress and structural relaxation. Predictions of this structural model are compared with experimental data on tempering and contrasted with pre- dictions of the viscoelastic model. Such comparisons revealed that, typically, structural relaxation accounts for approxi-

mately 24% of the total residual temper stresses.

I. Introduction

LASS scientists have long been interested in the role of G structural changes in the annealing and tempering of glass.’-3

Presented at the 79th Annual Meeting, The American Ceramic Society, Chicago, Illinois, April 27, 1977 (Glass Division, No. 47-G-77). Received May 19, 1977; revised copy received September 29, 1977.

‘Member, the American Ceramic Society.

The work of Adams and Williamson4 seemed to show that structural effects are negligible in annealing; later evidence refuted this con- c l ~ s i o n . ~ * ~ In tempering, the problem of the interaction of structural and stress relaxation was first raised by Acloque .’ The present paper is the first quantitative treatment of this subject.

Before 1960, progress in understanding processes such as anneal- ing and tempering was impeded by the lack of a mathematical framework and an accurate description of stress and structural relaxation in glass; these elements are now available. Lee et al. developed the mathematical framework for calculating stress gener- ation and Kurkjians and Narayanas~amy~ gave precise descriptions of stress and structural relaxation in glass. The theory of stress generation in glass by Lee er al. treated glass as a thermorheologi- cally simple viscoelastic material, thus incorporating stress relaxa- tion. Their formulation is referred to in this report as the “viscoelas- tic model.” Gardon and Narayanaswamy,6 in their analysis of annealing, used a “structural model” of stress generation that accounted for both stress and structural relaxation in glass. The structural model has been used in analyzing both annealing and tempering. lo,’’

This paper presents a theory of tempering that incorporates stress and structural relaxation and demonstrates the role of structural changes in tempering glass by comparing and contrasting computed