Journal of Non-Crystalline Solids 17 (1975) 147-157 © North-Holland Publishing Company

C A L C U L A T I O N O F B U L K M O D U L U S , S H E A R M O D U L U S A N D

P O I S S O N ' S R A T I O O F G L A S S

Akio MAKISH1MA and John D. MACKENZIE Materials Department, School of Engineering and Applied Science, University of California, Los Angeles, Cal., USA

Received 20 August 1974

Combining Griineisen's equation with our Young's modulus equation of glass, new formulae were semi-empirically derived for the calculation of bulk modulus, shear mod- ulus and Poisson's ratio of glass. Considering packing density of atoms and bond energy in unit volume the elastic moduli of glass can be calculated. The agreements between calculated and observed values of the moduli of glass axe satisfactory for more than 30 glasses.

I. In trod uc t ion

Bulk modulus, shear modulus and Poisson's ratio o f glass have been studied by many workers either to derive the relationship between chemical composition and these elastic moduli or to interpret the structure of glass.

Spinner [1] and Weir and Shartsis [2] measured shear moduli or bulk moduli of many optical glasses or alkali borate glasses by a dynamic method. Smyth [3] theo- retically studied the possible mechanisms involved in the elastic deformation of a silicate glass and concluded that Poisson's ratio should be 0.25 if the work done when glass is deformed consists in stretching or compressing the S i - O - S i links. Soga and Anderson [4] showed that the bulk modulus o f glass or crystal is related to the mean atomic volume and suggested that In K is inversely proportional to tn(2M/Po), where K is the bulk modulus, M is the molecular weight, P is the num- ber o f ions and p is the density.

The authors [5] theoretically derived an equation for the calculation o f Young's modulus o f glass considering the packing density o f atoms and the bond energy per unit volume. We further attempted to calculate the bulk modulus, shear modulus and Poisson's ratio of glass from its chemical composition using the same approach as in the previous work and found that combining Griineisen's equation, which de- fines the bulk modulus of solid materials, with our Young's modulus equation o f glass, the elastic moduli can be calculated from the chemical composition. In this paper, we report the derivation of the equations for the calculation of the bulk mo-

148 A. Makishima, J.D. Mackenzie~Calculation o f bulk modulus

dulus, shear modulus and Poisson's ratio of glass, and compare the calculated values with the observed values of the moduli of over 30 glasses.

2. Theory

According to Grtineisen's first rule [6], the bulk modulus K is represented as follows:

K V 0 = ~ mnlUoI , (1)

where V 0 is the equilibrium volume found where the net force vanishes, m and n are constants in Mie's equation, i.e.

¢ (r) = - A i r m + B/r n , (2)

and U 0 is the equilibrium energy for volume V 0 and is expressed for a ionic crystal with the interatomic spacing r 0 as follows:

U 0 = [(m - n)/n] (Nae2 / ro ) , (3)

where N is Avogadro's number, c~ is Madelung's constant and e is the electronic charge.

From eqs. (1) and (3) we obtain

K V 0 = m l m - n lN~e2 /9r 0 . (4)

The equilibrium volume, V0, is related to the interatomic spacing, r 0, by the equa- tion:

V 0 = CNr 3 , (5)

where C is a structural constant. From eqs. (4) and (5), the bulk modulus is calcu- lated as follows:

K = ( m l m - nl /9C) (e2 /ro)r~ 3 . (6)

The structural constant C is related to the packing density of ions as follows: for an oxide A x O y , radii of which are r 1 and r 2, the equilibrium volume, V0, is related to the interatomic spacing, ro, by

V 0 = CNr 3 = C N (r I + r2)3 . (7)

The packing density V t is calculated by the equation:

Vt= 4rrN(Xr ~ + y r3 ) /3Vo . (8)

A. Makishima, ZD. Mackenzie~Calculation of bulk modulus 149

From eqs. (7) and (8) we get

1/C = 3/3 Vt/47r, (9)

where

/3 = (r 1 + r2)3/(Xr~ + y r 3 ) . (10)

The/3 values of various oxides can be calculated using Pauling's ionic radii. We obtained an equation for the calculation of Young's modulus of glass, E, as

follows:

E = 2(e2/ro)ro 3 = 2V t G , (11)

where G is a dissociation energy per unit volume [5 ]. Combining eqs. (6), (9) and (11), we get the following equation:

K = (mlm - nl/24rr)/3 VtE . (12)

Eq. (12) shows that if we know m, n,/3, V t and Young's modulus of oxide materials, the bulk modulus can be calculated. However, for oxide glasses it is very difficult to evaluate m and n which are important factors of Mie's atomic potential-energy equa- tion. But eq. (12) indicates that the bulk modulus is correlated with the product of packing density and Young's modulus. We examined this correlation for more than 30 glasses by calculating the packing density of the glasses. We obtained a good line- arity between the observed bulk modulus and the product of the packing density and the observed Young's modulus of the glasses (see fig. 1). Therefore, the slope was determined to be 1.2 and the bulk modulus of glass can be expressed semi-em- pirically as follows:

K = 1.2 VtE= 2.4 v t 2 a . (13)

Shear modulus S and Poisson's ratio/a are shown by the following well-known equ- ations:

S-- 3EK/(9K - E) , (14)

p = E [ 2 S - 1. (15)

From these equations and eq. (13), we can get equations for the shear modulus and for Poisson's ratio. Thus eqs. (14) and (15) can be rewritten as follows:

S = [3.6 Vt/(10.8 V t - 1)1 E = 7.2 V2tG/(IO.S V t - 1), (16)

ta = 0.5 - 1/7.2 V t . (17)

These equations indicate that bulk modulus, shear modulus and Poisson's ratio of oxide glasses can be calculated if one knows the packing density and dissociation

150 A. Makishima, J.D. Mackenzie~Calculation of bulk modulus

"o o E

D ..Q

"17

> i_

t/1

0

1 2 0 0

I000

8 0 0

6 0 0

400

200

0 I i I I I I 0 200 400 600 800 1000 I z oo

V t E ( K bar ) Fig. 1. Agreement between observed values of bulk modulus and products of packing density and observed bulk moduli.

energy per unit volume o f oxide glasses. For the mult icomponent oxide glasses, Young's modulus was theoretically given in [5]:

E --- 83.6 V t ~ GiX i . (18)

This expression gives E in units of kbar if the unit o f G i, which is the dissociation energy of the i th oxide, is kcal/cm 3 and X i is the mole fraction of component i of an oxide glass. Therefore, eqs. (13) and (16) can be rewritten as follows:

K = 100 V2t ~ GiXi , (19)

S = [300 V2/(10.2 V t - 1)] ~ a i X i . (20)

3. Results and discussion

3.1. Silicate and other non-boron containing glasses

The bulk modulus, the shear modulus and Poisson's ratio of silicate and other

A. Makishima, J.D. Mackenzie~Calculation o f bulk modulus 151

I 0 0 0

L rO J31

"¢ 8 0 0

2~

"0 o 6 0 0 E

-5 ¢-I

4 O 0 - 0

>

.~ 200 0

[ I I I I

0 2 0 0 400 600 8 0 0 1000

C a l c u l a t e d bulk m o d o l u s ( K b a r )

Fig. 2. Agreement between observed values of bulk modulus and those calculated using eq. (19).

non-boron containing glasses were calculated using eqs. (17), (19) and (20) and the results are compared with observed values (see figs. 2, 3 and 4). Figs. 2, 3 and 4 show a fairly good agreement between observed and calculated values. The straight lines in these figures have a gradient of unity. The glass compositions used in these calculations are given in table 1. These glasses are different from those used in fig. 1. The observed bulk moduli, Poisson's ratios and densities of the glasses are shown in table 2. Packing density V t and dissociation energy per unit volume G i were calculated by the method reported by the authors in [5].

3.2. Borate glasses

It is necessary to take into consideration the change of the coordination number of boron in borate glasses. Using the method described in our previous paper [5], we calculated the bulk modulus and shear modulus of some borate glasses. If we know the fraction ~/of BO 4 tetrahedra, two dissociation energies per unit volume, one for BO 3 and one for BO4, can be used for the calculation of elastic moduli. For the calculation of G i in eqs. (19) and (20), the G i should be

G B = 3, (G 4 - G3) + G3, (21)

where G 4 = dissociation energy per unit volume for BO 4 = 18.6 kcal/cm 3 and

152 A. Makishima, J.D. Mackenzie~Calculation o f bulk modulus

"E q~ n

D

D "O

o

E

-O

o~ b~

O

6 0 0

5 0 0

4 0 0

3 0 0

2 0 0

100

0 0

I I I I I I 100 200 300 400 500 600

C a l c u l a t e d s h e a r m o d u l u s ( K b a r )

Fig. 3. Agreement between observed values of shear modulus and those calculated using eq. (20).

0.3 _o + -

o • •

• : - .

0.2

0

0 . 1 L I i I

0.l 0.2 0.3

C a l c u l a t e d Poisson's ra~rio

Fig. 4. Agreement between observed values of Poisson's ratio and those calculated using eq. (17).

Tab

le 1

. C

ompo

siti

ons

of

glas

ses

used

in

the

pres

ent

calc

ulat

ion.

Gla

ss

Co

mp

on

ent

oxid

es (

mol

e%)

no.

(mol

%)

Si0

2 N

a20

K2

0

Li2

0 A

12

03

C

aO

PbO

B

aO

ZnO

B

eO

Oth

er o

xide

s (m

ol%

)

1 76

.8

0.6

8.7

2 72

3.

2 8.

4 3

69.6

7.

3 5.

6 4

71.1

4.

5 5.

6 5

69.2

1.

1 6.

9 6

68.2

6.

6 7

66.8

5.

3 8

64.6

3.

8 9

61.7

2.

9 10

68

.9

2.0

7.2

11

66.8

7.

3 12

7

Z7

6.

5 9.

7 13

40

.7

14

100

15

48.0

16

37

.2

17

75

18

70

19

65

20

11.6

21

47

.3

22

52.8

23

50

30

.0

24

55

25.0

25

62

.5

17.5

26

47

.5

27.5

27

52

.5

22.5

28

57

.5

17.5

29

49

.0

~30

54.9

P

25

30

35

7.2

5.7

6.6

28.3

11

.2

5.7

6.2

6.5

13.6

0.

3 16

.4

17.5

18

.8

22.8

25

.2

27.9

31

.6

35.4

6.

9 7.

1 9.

5 7.

7 3.

1

10.3

52.4

11.8

7.7

7.9

8.7

3.0

25.4

27.0

31

.8

22.8

20

.3

t

21.8

13

.3

26.7

(M

gO)

7.7

(MgO

), 1

.4 (

ZrO

2)

24.4

(M

gO)

14.1

(M

gO),

4.6

2 (Z

rO2)

21

.2 (

MgO

) 20

.0 (

TiO

2)

20.0

(T

iO2)

20

.0 (

TiO

2)

25.0

(T

iO2)

25

.0 (

TiO

2)

25.0

(T

iO2)

23

.0 (

MgO

) 8.

2 (M

gO),

5.4

(Z

rO2)

av

E"

s,

154 A. Makishima, £D. Mackenzie/Calculation o f bulk modulus

Table 2 Observed bulk moduli, shear moduli, Poisson's ratios, densities and calculated packing densities of various glasses.

Glass Bulk Shear Poisson's density Packing Ref. no. modulus modulus ratio density

(kbar) (kbar)

1 340 226 0.228 3.227 0.529 [ 1 ] 2 352 224 0.238 3.446 0.531 [ 1 ] 3 344 231 0.227 3.547 0.530 [ 1 ] 4 323 232 0.211 3.605 0.531 [1] 5 306 222 0.215 3.860 0.529 [1] 6 298 223 0.200 4.012 0.530 [1] 7 326 223 0.213 4.215 0.533 [1] 8 332 221 0.228 4.485 0.532 [1] 9 334 220 0.230 4.767 0.535 [1]

10 386 261 0.224 3.288 0.540 [1] 11 374 256 0.221 3.474 0.538 [1] 12 415 256 0.241 2.702 0.540 [1] 13 989 547 0.267 2.787 0.630 [7] 14 361 314 0.162 2.20 0.514 [1] 15 824 483 0.256 2.723 0.615 [7] 16 1005 545 0.270 2.785 0.640 [7] 17 494 317 0.236 2.302 0.548 [8] 18 509 317 0.242 2.334 0.556 [8] 19 498 331 0.228 2.354 0.562 [8] 20 798 420 0.276 2.051 0.635 [8] 21 813 504 0.243 2.781 0.581 [8] 22 742 479 0.234 2.629 0.598 [8] 23 500 302 0.249 2.749 0.556 [9] 24 501 314 0.241 2.740 0.566 [9] 25 494 334 0.224 2.703 0.571 [9] 26 526 315 0.250 2.811 0.576 [9] 27 530 335 0.239 2.800 0.579 [9] 28 523 345 0.230 2.771 0.581 [9] 29 876 500 0.260 2.681 0.616 [7] 30 744 437 0.255 2.795 0.645 [7]

G 3 = dissociation energy per uni t volume for BO 3 = 3.9 kcal /cm 3, 7 is the fraction of BO 4 tetrahedra in the glass. This equat ion was used for the calculation of Young's

modulus o f borate glasses. The measured and calculated bulk modul i and shear modul i o f some borate glasses

are tabula ted in table 3 and plotted in fig. 5. They show comparatively good agree- ment considering the uncer ta in ty o f experimental data and the approximat ion underlying eqs. (19) and (20). The straight line in fig. 5 has a gradient o f uni ty .

3.3. Discussion

It is interest ing to note that the shear modulus is part ly expressed by the square

A. Makishima, J.D. Mackenzie~Calculation o f bulk modulus 155

Table 3 Comparison of the measured and calculated bulk moduli and shear moduli of some borate glasses.

Borate glass Bulk modulus (kbar) Shear modulus (kbar) (mol%) Measured [8] Calculated Measured [8] Calculated

100B20 121 105 69 71 5.6Na20-94.4B203 190 160 110 97 11Na20-89B203 250 210 130 123 17Na20-83B203 290 250 160 145 17Li20-83B203 330 336 190 191 17K20-83B203 240 213 120 130 17Cs20-83B203 200 182 90 118 22Na20-78B2Oa 350 300 180 172

of the packing density, and Young's modulus is correlated linearly with the packing density as shown in eqs. (18) and (19). The bulk modulus is defined as the three- dimensional isotropic pressure divided by the relative volume change. Therefore, the bulk modulus would be more sensitive to the packing density. The shear modu- lus is related to the packing density in a complicated way shown in eq. (20), but for oxide glasses Poisson's ratio is determined by the packing density as shown in eq. (17). If atoms are loosely packed in the oxide glass, Poisson's ratio is small whereas tightly packed glass has a higher Poisson's ratio. This seems reasonable because Poisson's ratio is defined by the ratio of transverse to axial strain, and the trans- verse strain would be smaller in loosely packed glass as there is more space for the atoms to move in.

400

"2 300

~ 2oo

} IO0

o

• Butk m o d u l u s

oShear modulus +. . /

o<

i i i i 0 0 100 200 300 400

C a l c u l a t e d e l a s t i c m o d u l u s ( K b a r )

Fig. 5. Agreement between observed and calculated bulk moduli and shear moduli of some bo- rate glasses [8].

156 A. Makishima, J.D. Mackenzie/Calculation o f bulk modulus

From eqs. (12), (14) and (15) Poisson's ratio and the shear modulus can be theoretically expressed as follows:

S = [mlm - nl ~ V t / ( 3 m l m - nil3 V t - 8zr)] E , (22)

/a = 0.5 - 4~r/mlm - n l~ V t . (23)

These equations show that Poisson's ratio as well as the shear modulus are correl- ated via a complicated expression in m and n which are important factors in Mie's atomic potential-energy equation.

Shannon et al. [10] reported the most recently compiled atomic radii, and using their atomic radii, we can get nearly equal or bigger packing densities than those calculated using Pauling's atomic radii for glass compositions in table 1. Therefore, a slightly better agreement between observed and calculated elastic moduli is ob- tained. In this paper we used Pauling's radii which are well-known.

3.4. App l i ca t ions

The results of the present study can be used to predict the chemical composition of glasses with the highest possible bulk modulus and shear modulus.

The highest Young's modulus of oxide glass calculated would be closer to 16 × 102 kbar, assuming that the packing density is about 65% and Y~ G i X i = 27.5 kcal/cm 3. Eqs. (19) and (30) show that the highest bulk modulus and shear modu- lus would be 11.6 × 102 kbar and 619 kbar, respectively, using the same assump- tion in the prediction of the highest Young's modulus of oxide glass, whose main oxide constituents are BeO, AI203 and ZrO 2.

According to Laberge et al. [ 11 ] the density fluctuation in oxide glass, which correlates with the intrinsic optical loss in fiber-glass wave guides for long-range communication, can be expressed as follows:

Ap21p 2 = k T f l V K T , (24)

where k is Boltzmann's constant, Tf is the fictive temperature, K T is the bulk modu- lus and V is the scattering volume. Eq. (24) shows that a glass composition with a lower fictive temperature and a high bulk modulus has a low density fluctuation. Laberge et al. [11] reported that 50% Li20-B203 glass may be a suitable choice for making optical fiber. This seems reasonable because, among alkali-borate glasses which have comparatively low liquidus temperature, lithia borate glass has a higher bulk modulus as the dissociation energy per unit volume of Li20 is fairly large (19.2 kcal/cm 3) compared with that of Na20 (8.9 kcal/cm 3) or K20 (5.6 kcal/cm3), and hence lowers the scattering as indicated by eq. (24).

A. Makishima, J.D. Mackenzie/Calculation of bulk modulus 157

Acknowledgements

This work has been supported by the Directorate of Chemical Science, AFOSR, under Grant No. 70-1856 for which we are most grateful.

The authors are greatly indebted to N. Soga for constructive criticism of this paper.

References

[1] S. Spinner, J. Amer. Ceram. Soc. 37 (1954) 229. [2] C.E. Weir and L. Shartsis, J. Amer. Ceram. Soc. 38 (1955) 299. [3] H.T. Smyth, J. Amer. Ceram. Soc. 43 (1959) 276. [4] N. Soga and O.L. Anderson, Proc. 7th Intern. Congr. on Glass, Brussels, Belgium, 1965,

vol. 1 (Institut National du Verre, Charleroi, Belgium, 1966) p. 9. [5] A. Makishima and J.D. Mackenzie, J. Non-Crystalline Solids 12 (1973) 35. [6] C. Zwikker, Physical Properties of Solid Materials (Interscience, New York, 1954) p. 90. [7] M.H. Manghnani, Elastic Parameters and Infrared Characters of Glasses under High Pres-

sures and Temperatures, Final Report, Contract N00014-67-A-0387-0005, NR 032-515 (1970).

[8] R.R. Shaw and D.R. Uhlmann, J. Non-Crystalline Solids 5 (1971) 237. [9] M.H. Manghnani, J. Amer. Ceram. Soc. 55 (1972) 360.

[10] R.D. Shannon and C.T. Prewitt, Acta Cryst. B25 (1969) 925. [11] N.L. Laberge, P.K. Gupta, V. Vasilescu and P.B. Macedo, J. Non-Crystalline Solids 13

(1974) 164.