Direct Calculation of Young's Modulus of Glass

8/18/2019 Direct Calculation of Young's Modulus of Glass

1/11

Journal of Non-Crystalline Solids,

12 (1973) 35 45. © North-Holland Publishing Co m pan y

D I R E C T C A L C U L A T I O N O F Y O U N G ' S M O D U L U S O F G L A S S

A . M A K I S H I M A an d J .D . M A C K E N Z I E

Materials Department, School o f Engineering and Applied Science,

University of California. Los Angeles. USA

Received 16 Novem ber 1972

An equation has been derived for the d irect calculation of the Y oung's modulus o f oxide

glasses from their chemical composit ions. The me thod is based on a consideration of dissociation

energy of the oxide consti tuents per unit volume and the packing density. F or borate glasses,

the ratio o f four-coordinated to three-coordinated boro ns must be taken in to consideration.

Excellent agreement is obtained between calculated and measured values of Yo ung 's modulus

for over thirty different glasses.

1 . I n t r o d u c t i o n

Y o u n g ' s m o d u l u s o f g la s s h a s b e e n e m p i r i c a l l y s tu d i e d b y m a n y a u t h o r s e i t h e r

t o d e ri ve t h e r e l at io n s h ip b e t w e e n c h e m i c a l c o m p o s i t i o n a n d Y o u n g ' s m o d u l u s o r

t o o b t a i n h i g h Y o u n g ' s m o d u l u s g la ss c o m p o s i t i o n s f o r t h e m a n u f a c t u r e o f s tr o n g

g lass f ibe r s . F o r ex am ple , C app s e t a l. [ 1] i n v e s ti g a te d Y o u n g ' s m o d u l u s o f gla ss

w i t h a w i d e r a n g e o f c h e m i c a l c o m p o s i t i o n s , a n d s u c c e e d e d i n d e v e l o p i n g a gla ss

w i t h t h e l l ig h e s t Y o u n g ' s m o d u l u s v a l u e ev e r o b t a i n e d , 1 4 4 8 k b a r . P h il li p s [ 2 ]

p r o p o s e d a n e m p i r ic a l m e t h o d t o p r e d ic t t h e Y o u n g ' s m o d u l u s f r o m c h e m i c a l

c o m p o s i t i o n . W i ll ia m s a n d S c o t t [ 3 ] s u b s e q u e n t l y o b t a i n e d m o r e e x a c t v a lu e s o f

t h e o x i d e c o e f f i c ie n t s , a g a i n f o r e m p i r i c a ll y c a l c u l a t i n g Y o u n g ' s m o d u l u s o f a lk a li -

f ree g lasses.

Y o u n g ' s m o d u l u s h a s a l s o b e e n s t u d i e d i n s o m e w h a t m o r e t h e o r e t i c al t e rm s .

C h a r le s [ 4 ] s u g g e s t e d t h a t t h e d e n s i t y o f th e S i - O - S i b r id g e s i n g la ss g o v e r n s th e

Y o u n g ' s m o d u l u s . L o w e n s t e in [ 5 ] , in s t u d y i n g th e r o le p la y e d b y o x i d e s a d d e d t o

a b a s e g l a s s , p o i n t e d o u t t h a t o x i d e s w i t h h i g h f i e l d s t r e n g t h s h a v e a t e n d e n c y t o

e n h a n c e t h e Y o u n g ' s m o d u l u s v a lu e s . G i l m a n [ 6 ] s h o w e d t h a t i n g e n e r a l t h e e la s ti c

m o d u l u s o f m a t e ri a ls i s r e l a te d t o t h e s e p a r a t i o n d i s t a n c e o f a t o m s , a n d i n v e rs e l y

p r o p o r t i o n a l t o t h e f o u r t h p o w e r o f a t o m i c s p ac in g s .

A t p r e s e n t, n o t h e o r y d e s c r i b e s th e d e p e n d e n c e o f Y o u n g ' s m o d u l u s o f g la s s o n

e i t h e r s t r u c t u r e s o r c h e m i c a l c o m p o s i t i o n s . I n t h is p a p e r , w e re p o r t t h e d e v e l o p m e n t

o f a m o d e l f o r t h e t h e o r e t i c a l c a l c u l a t i o n o f Y o u n g ' s m o d u l u s o f gla ss es f r o m t h e ir

c h e m i c a l c o m p o s i t i o n s .


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36 A. M ak ishima, J .D. Mackenz ie, You ng s m odu lus o f g lass

2 . T h e o r y

Accord ing t o G i lman [6 ] , t he Yo ung ' s mo du lus (E ) o f ion i c c rys ta l s can be ap -

p rox im a te ly de r ived a s fo l l ows .

Fo r a pa i r o f i ons o f op pos i t e s ign wi th t he spac ing r 0 , t he e l ec t ro s t a t i c ene rgy

of a t t r ac t i on U is equa l t o

U = - e 2 / r O . (1 )

In o rde r t o acco un t f o r t he ma ny in t e r ac t i ons be tw een ions w i th in a c rys t a l, t h i s is

mu l t ip l i ed by t he Made lung cons t an t a , g iv ing t he M ade lung ene rgy :

U m = a U . (2 )

The fo r ce be tween ions is aUra~Or, and so the stress o is

o ~ l i ~ ) U m ~

r 2

(3 )

Then the change o f st r es s fo r a change i n r is do /d r , and t he r e fo re

d o r ~ \ 3 r 2 ]

(4)

bu t th i s is jus t E de w here the s t ra in de = dr / r O . Thus ,

d o 1 / 0 2 U r n I 2 a e 2

E = - - ~ - - ~ ( 5 )

de r 0 \ 3r 2 ] r 4

Eq . ( 5 ) shows tha t t he Y oung ' s m odu lus o f i on i c c rys ta l s i s i nve r se ly p rop or t i on a l

t o t he fou r th pow er o f a tomic spac ings r 0 , and t h i s re l a t i onsh ip was con f i rm ed by

m any ion i c and even cova l en t c rys ta l s . Th i s r e la t i on has no t been eva lua t ed fo r

glasses.

We can now rewr i te eq . (5) as fo l lows:

E = 2°t ( e2 )

r 3 70-0 = 2 ° ~ 0 (6)

F ro m eq . ( 6 ) and eq . ( 2 ) , t he Y oung ' s mo du lus i s two t imes t he b ind ing ene rgy

Um per c ubic vo lum e o f r03.

The sing le bon d s t r eng th o f ox ides has been de t e rm ined by Sun [7 ] f rom the

ra t i o o f t he d i s soc i a t ion ene rgy and t he coo rd ina t i o n num ber , and such d i s soc ia t i on

ene rg i e s (U ' ) o f va r ious ox ides a r e shown in t ab l e 1 . F ro m eq . ( 6 ) and t he m o la r


3/11

A. M ak i sh im a , J .D . M ac k e nz ie , Y oung s m odu lus o f g la s s

Table 1

Dissociation energies per mole and per unit volume and packing factor of oxides.

37

Oxide U' (kcal/mole) G (kcal/cm 3) V

Al2 03 804 32 21.4

BeO 250 30.0 7.0

ZrO2 485 23.2 15.1

TiO2 435 20.7 14.6

Sc203 724 20.2 23.6

MgO 222 20.0 7.6

ThO2 516 19.3 28.5

Li20 288 19.2 8.0

B20 3 712 18.6 20.8

Y20 3 798 17.7 24.8

Ga203 534 17.1 21.9

La20 3 812 16.2 28.4

CaO 257 15.5 9.4

SiO2 424 15.4 14.0

P205 884 15.0 34.8

As2Os 698 13.1 36.2

SnO2 278 12.9 17.4

SrO 256 11.6 10.5

ZnO 144 9.9 7 . 9

BaO 260 9.7 13.1

PbO 2 232 9.1 15.3

Na20 240 8.9 11.2

CdO 119 7.6 9.2

ln203 259 6.7 23.5

K20 230 5.6 18.8

Pb20 230 4.5 9.9

PbO t45 4.2 11.7

Cs20 228 3.4 31.2

volumes, we calculated the dissociation energy per unit volume (G) of oxides, which

are also tabulated in table 1.

If the A-O bond energy in one molecule of oxide A x O Y is similar for the crys-

tal and the glass, providing the c oord inati on num ber is the same, the n it is reasonable

to apply the above treatment to oxide glasses. However, because of the disordered

structure of glass, it is difficult to adopt a meaning ful Madelung con stan t as for

crystalline oxides. In place of Um per cubic vo lume r03 , we consider that a more

approp riate bin din g energy for glass (Urn) is given by the produce of the d issociati on

energy per uni t volume (G) and the packing density of ions, V .

For example, in a simple one-component glass such as fused silica,

E = 2 V t G . (7)


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38 A. M ak i sh im a , J .D . M ac k e nz ie , Y oung s m odu lus o f g la s s

Table 2

O bse r ve d Y oun g ' s m odu lus , de ns i t y , c a l c u l a te d pa c k ing de ns i ty a nd b ind ing e ne r gy o f va rious

glasses.

G las s num be r O bse r ve d D e ns i ty P a c k ing B ind ing R e f .

Y ou ng ' s m od u- ( g / c m 3 ) de ns i ty V e ne r gy U '~ ,

l us E ( k b a r) ( k c a l / c m )

1 827 2 .428 0.559 10.1 [9] (e)

2 802 2 .384 0 .552 9 .66 [9] ( e )

3 791 2.358 0 .545 9.26 [9] (e)

4 763 2.335 0 .54 2 9.10 [9] (e)

5 762 2 .319 0 .537 8 .91 [9] ( e )

6 1093.5 2.996 0 .567 11.5 [9] (b)

7 1096 .2 3.170 0.56 0 11.9 [91 (b)

8 742 2 .231 0 .525 8 .35 [10]

9 764 2 .258 0 .532 8 .62 [10]

10 769 2 .277 0 .538 8 .81 [10]

i1 784 .4 2 .30 0 .548 9 .07 [10]

12 788.1 2 .334 0 .556 9 .28 [10]

13 754.5 2 .749 0 .566 8 .24 [91 (d)

14 779.5 2.74 0 0.571 8.48 [91 (d)

15 81 8.0 2.703 0 .57 6 8.81 [91 (d)

16 789 .8 2.811 0 .576 8.57 [9] (d)

17 830 .7 2 .800 0 .579 8 .87 [9] (d)

18 847.5 2.771 0 .581 9.07 [91 (d)

19 412.1 7 .158 0 .534 4 .75 [101

20 435 6 .676 0 .530 5 .67 [10]

21 493 6 .35 0 .541 6 .38 [10]

22 73 0 2.20 0 .514 7.95 [9] (a)

23 111 0 3.09 0 .562 12.7 [9] (c)

24 115 0 3.19 0 .593 13.5 [9] (c)

25 890 2 .794 0 .583 9 .9 [91 (c )

26 79 0 2.81 0 .59 3 8.9 [9] (c)

27 55 5 3.22 0.521 6.7 [91 (a)

28 535 4.01 0 .539 6.31 [9] (a)

29 136 8 2.77 0 .63 13.6 [1]

30 1448 2.815 0 .63 14.9 11]

F o r p o l y c o m p o n e n t g la ss es ,

E = 2 V t ~ G i X .

i

T h e p a c k i n g d e n s i t y V i s d e f i n e d b y ,

P ~ V i X i

t = ~

i

( 8 )

( 9 )


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A . M a k i s h im a , J . D . M a c k e n z i e , Y o u n g s m o d u l u s o f g la ss

Table 3

Compositions of glasses used in the present calculation

39

Gl as s Component oxides (Mole )

No.

Si O2 A 1 2 0 3 Li20 Na20 MgO CaO Other oxides

1 66.9

2 75.2

3 80.2

4 83.4

5 85.8

6 7

7

8 9{}

9 85

10 80

11 75

12 70

13 50.0

14 55.0

15 62.5

16 47.5

17 52.5

18 57.5

19 45

20 50

21 55

22 100

23

24

16.7 16.4

12.5 12.3

10.0 9.8

8.4 8.2

7.2 7.0

28

34

10

15

20

25

30

30.0

25.0

17.5

27.5

22.5

17.5

7 57

9 50 7 (BaO)

20 (TiO2)

20 (TiO2)

20 (TiO2)

25 (TiO2 )

25 (TiO2)

25 (TiO2)

65 (PbO)

50 (PbO)

45 (PbO)

41.5 6.8 8.0 28.6 4.7 (BaO), 3.3 (TiO2),

3.4 (ZrO2)

41.6 3.4 4.95 30.0 4.95 (BaO), 4.95 (TiO2),

8.35 (ZrO2), 1.65

(K20)

25 49.8 5.1 19.8 25.0

26 93.5 0.3 6.3 (BaO)

27 76.8 0.6 8.7 (K20), 13.6 (PbO)

28 68.2 6.6 (K20), 25.2 (PbO)

29 40.69 7.19 26.70 25.42 (BeO)

30 26.16 17.99 28.61 27.24 (BeO)

where M is the effecti ve molecu lar weight, p is the density, X i is the mole fraction of

com pon ent i, and F i is a packing factor obtai ned fr om the fo llowing equatio n for

an oxide AxOy:

V

=6 .0 23 X 10 23

~ T r ( X R 3 + y R 3 0 ).

(lO)

Values of V i are shown in table 1. R A and R O are the respective ionic radius of


6/11

4 0 A. Makishima, J.D.

M ackenz ie , Youn g s modu lus of g lass

14

0

- I Z

X

I 0

v

: 3

J

8

0

~ 6

Z

0

4

.I

t~

y

m ~ I I I I I I

2 4 6 8 I 0 1 2 1 4

C A L C U L A T E D ¥ O U N G S M O D U L U S ( K b a r ) X I O - 2

Fig . 1 . A g r e e m e n t b e t w e e n m e a s u r e d v a l u e s o f Y o u n g ' s m o d u l u s a n d t h a t c a l c u l a t e d f r o m e q . ( 1 1 ) .

m e t a l a n d o x y g e n . ( I n t h e p r e s e n t st u d y , P a u l i n g 's i o n i c r a d i i [ 8 ] a re u s e d . )

T h u s , t h e Y o u n g ' s m o d u l u s o f g l a s s i s t h e o r e t i c a l l y g i v e n b y

E = 8 3 . 6 V

~ a i X i .

( 1 1 )

i

T h i s e x p r e s s i o n g iv e s E i n u n i t s o f k b a r i f u n i t s o f G a r e i n k c a l / c m 3 .

3 . R e s u l t s a n d d i s c u s s io n s

3 . 1 . S i l i c a t e and o th e r non -bo ron con tai n i ng g l asses

T h e c a l c u l a t e d v a l u e s o f binding e n e r g y ( U m ) ' a n d p a c k i n g d e n s i t y o f v a r i o u s

g la ss es ar e s h o w n i n t a b le 2 t o g e t h e r w i t h t h e m e a s u r e d d e n s i ty a n d v a lu e s o f E o b -


7/11

A. Makishima, J.D. Mackenzie, Young s modulus of glass 41

1 1 4 0

I 1 0 0

8

1 0 6 0

13

v

- -

1020

D

0

o - 9 8 0

c

o

)-

9 4 0

C d 0

• Z n O

N a 2 0 . - ',

. -

•

S r O

. . - 8 a O

K 2 0

Z r O 2 . . - ,

• B e O

L i O 2 . -

• T i 0 2 . - -

T h ? . 2 .

C a O . - •

.- - M 9 0

L

0 5

I0 15

2 0 2 5 3 0

D i s s o c , d t , o n E n e r g t3 ( K c a l / ~ c )

Fig. 2. Relationship between measured Young's m odulus and the dissociation energy per unit

volume of various oxides which are m inor addit ions to the b ase glass [5 ].

t a i ned by va r i ous au t ho r s u s i ng t he son i c m e t hod . T he com pos i t i ons o f t he se g l a s se s

a re show n i n t ab l e 3 . In fi g. 1 , t he m easu red Y o ung ' s m o du l i o f t he se 30 g la sses a r e

p l o t t e d aga i n s t va l ues ca l cu l a t ed f ro m eq . (11 ) . T he so li d s t r a i gh t li ne r ep re sen t s a

s l ope o f un i t y . C ons ide ring~the unc e r t a i n t y o f exp e r i m en t a l da t a and t he ap p rox i -

m a t i ons und e r l y i ng eq . (11 ) , t he a g reem en t i n f ig . 1 i s cons i de red t o be ve ry sat is -

f a c t o r y .

L o w e n s t e i n [ 5] s t u d ie d t h e e f f e c t s o f e qu a l a m o u n t s o f v a r i o u s o x i d e s o n t h e

Y o u n g ' s m o d u l u s o f a b a se g la ss o f c o m p l e x c o m p o s i t io n . W e h a v e p l o t te d t h e d is -

s o c i at io n e n e r g y p e r u n i t v o l u m e o f o x i d e ( G ) ag a in s t t h e o b s e r v e d Y o u n g ' s m o d u -

lus of such g lasses . In f ig . 2 , t he re l a t ionship be tween E and G i s seen to g ive an ap-

p r o x i m a t e l y l in e a r p l o t w h i ch v e r if ie s t h e a b o v e t r e a t m e n t o f E .

P h i ll ip s [2 ] ha s em p i r i ca l l y ob t a i n ed c oe f f i c i en t s fo r t he ca l cu l a t ion o f t he

Y o ung ' s m o du l us o f g la ss f ro m chem i ca l com pos i t i on . W illiam s and S co t t [3]

r ecen t l y r e f i ned t he se em p i r i ca l coe f f i c i en t s fo r a l ka l i- f r ee g la sses . S uch coe f f i c i en t s

a re no t con s t an t bu t co u l d va ry g rea t l y acco rd i ng t o t he g l as se s in ques t i on . W e

have p l o t t ed t he i r coe f f i c i en t s aga in s t the d i s soc i a t ion ene rgy o f ox i de pe r un i t

v o l u m e ( G ) , a n d f o u n d t h a t t h e c o e f f i c ie n t s ar e n o t e x a c t l y r e l a te d l in e a r l y t o ( G )


8/11

4 2

A . M aki shima, , I.D , Mackenzie, Young s modulus f g las s

r ~

o ~

3 0

0

E

x~ 2 5

- t -

g

.u_ 2 O

0

m 1 5

D

0

0

> -

• P h i l l i p s

0 W i l l i a m s a n d S c o t t

d ~

0

p -

Z reO .

C a O j - e T i 0 2 B e O

I

/ s

' i

b O

/ I I I I l I

15

A I 2 0 3

0

0 5 I 0 2 0 2 5 3 0 3 5

D i s s o c i a t i o n E n e r g y ( K c a l T c c )

Fig . 3 . P l o t o f e m p i r i c a l Y o u n g ' s m o d u l u s c o e f f i c i e n t a n d d i s s o c i a t io n e n e r gy o f o x i d e s .

a s s h o w n i n f i g . 3 . A l u m i n i u m o x i d e i s a c l e a r e x c e p t i o n . I t i s a p p a r e n t t h a t t h e

e m p i r i c a l ly o b t a i n e d c o e f f i c i e n t s c a n n o t b e s i m p l y r e l a te d t o t h e d i s s o c i a t i o n

e n e r g y p e r u n i t v o l u m e o x i d e s .

3 . 2. B o r a t e

glasses

I t i s w e l l - k n o w n t h at t h e c o o r d i n a t i o n o f b o r o n i n g la ss v a r ie s f r o m 3 t o 4 . T h i s

f a c t m u s t o b v i o u s l y b e t a k e n i n t o c o n s i d e r a t i o n f o r t h e c a l c u l a t i o n o r E . T h e v a l u e

o f G f o r B 2 0 3 i n ta b le 1 i s se e n t o b e f a ir l y h i g h , b u t t h e Y o u n g ' s m o d u l u s o f b o r i c

o x i d e g la ss i s a p p a r e n t l y v e r y l o w . T h i s m a y b e a t t r ib u t e d t o t h e s t r u c t u r e o f B 2 0 3

T h e l o w Y o u n g ' s m o d u l u s o f b o r i c o x i d e i s p r o b a b l y c a u s e d b y t h e w e a k b i n d i n g

f o r c e s b e t w e e n p l a n e s o f i n t e r l i n k i n g B O 3 t r ia n g l e s. T h i s i s s o m e w h a t s im i l a r t o

t h e c a se o f g r ap h i te . F r o m e q . ( 1 1 ) w e c a l c u la t e d G f o r b o r ic o x i d e g la ss t o b e 3 . 9

k c a l / c m 3 u s in g t h e m e a s u r e d v a lu e o f Y o u n g ' s m o d u l u s o f 1 7 0 k b a r [ 1 0 ] . T h i s i s

m u c h l o w e r t h a n t h e v a lu e o f 1 8 . 6 k c a l / c m 3 i n ta b l e 1 . T h i s d i f f e r e n c e i s t h u s d u e

t o t h e t w o - d i m e n s i o n a l , i . e ., s h e e t -l i k e s t r u c tu r e , a n d t h r e e - d i m e n s i o n a l l i n k i n g o f

B - O b o n d s .

I f w e k n o w t h e f r a c t i o n 3 ' o f B O 4 t e t r ah e d r a i n t h e g l a ss , t w o d i s s o c i a t i o n e n e r -

g i es p e r u n i t v o l u m e , o n e f o r B O 3 a n d o n e f o r B O 4 , c a n n o w b e u s e d f o r t h e c al-


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A. M a k ish i ma, J . D . M ac k e nz ie , Young s mod ul us o f g las s

Table 4

Com parison of measured and calculated Y oun g's modulus of so m e borate glasses.

43

Borate glasses

(mole %)

Young's modulus (kbar)

Measured [ 10]

Calculated

5 .6 N a2 0-9 4 .4 BzO 3 260 240

11 N a2 0- 89 B203 340 305

17 N azO -83 B203 390 364

17 L i2 0- 83 BzO 3 480 480

1 7 K 2 0 - 8 3 B 2 0 3 3 00 3 25

17 C s2 0-8 3 B203 250 290

2 2 N a 2 0 - 7 8 B 2 0 3 4 7 0 4 30

c u l a ti o n o f Y o u n g ' s m o d u l u s . T h u s , G i i n e q . ( 1 1 ) s h o u l d b e s u b s t i t u t e d w i t h a

m o d i f i e d e n e r g y t e r m , G B :

G B = 3 ' ( ( ; 4 - G 3 ) + G 3 ,

( 1 2 )

w h e r e G 4 - - d i s s o c i a t i o n e n e r g y p e r u n i t v o l u m e = 1 8 . 6 k c a l / c m 3 a n d G 3 = di s-

s o c i a t i o n e n e r g y p e r u n i t v o l u m e = 3 . 9 k c a l / c m 3 . S u c h a c a l c u l a ti o n o f E f o r a bi -

n a r y b o r a t e g la ss is s h o w n b e l o w . A c c o r d i n g t o S h a w e t ah [ 1 0 ] t h e o b s e r v e d

Y o u n g ' s m o d u l u s f o r a

n o n - p h a s e s e p a r a t e d

g la ss o f th e c o m p o s i t i o n 7 2 . 5 B 2 0 3 -

2 7 . 5 N a 2 0 ( in m o l e % ) is a b o u t 5 6 0 k b a r . A c c o r d i n g t o B r a y [ 11 ] , 7 f o r th i s g la s s

is a p p r o x i m a t e l y 0 . 4 . F r o m e q s . (1 I ) a n d ( 1 2 ) a n d u s i n g th e a b o v e v a l u e s o f G 4

a n d G 3 , E i s c a l c u l a t e d t o b e 5 2 0 k b a r . C o n s i d e r in g t h e u n c e r t a i n t y i n 7 , a g r e e m e n t

b e t w e e n t h e o b s e r v e d a n d c a l c u l a t e d v a l u e s is s a ti s f a c t o ry . O b v i o u s l y , e q . ( 1 2 ) c a n

b e u s e d t o e s t i m a t e 7 i f E i s k n o w n . I n t h e c a s e o f th e a b o v e s o d i u m b o r a t e g la ss ,

t he ca l c u la t e d )' i s 0 .43 .

T h e m e a s u r e d a n d c a l c u l a t e d Y o u n g ' s m o d u l i o f o t h e r b o r a t e g l as se s a r e t a b u la -

t e d i n t a b le 4 , a n d a g a i n s h o w s c o m p a r a t i v e l y g o o d a g r e e m e n t . T h e v a lu e s o f 7 u s e d

a r e t h o s e r e p o r t e d b y B r a y [ 1 1 ] . F o r t h e s e g l a s s e s , t h e s l i g h t d i s c r e p a n c y c a n a l s o

b e t h e r e s u lt o f p h a s e - s e p a r a t io n i n t h e g la ss . T h e m e a s u r e d Y o u n g ' s m o d u l i o f

v a r i o u s g la ss es c o n t a i n i n g b o r i c o x i d e a r e p l o t t e d a s a f u n c t i o n o f m o l e p e r c e n t o f

B 2 0 3 , i n fi g. 4 . I t s h o w s t h a t s m a ll a d d i t i o n o f b o r i c o x i d e t o t h e t w o b a s e g l a ss e s

i nc re a se s t h e Y o u n g ' s m o d u l u s . T h i s m a y b e a t t r ib u t e d t o t h e f o r m a t i o n o f B O 4

t e t r a h e d r a . O n t h e o t h e r h a n d , t h e a d d i t i o n o f s m a l l a m o u n t s o f o t h e r o x i d e s s u c h

a s P b O a n d N a 2 0 t o b o r i c o x i d e a ls o i n c re a s e s th e m o d u l u s . I n a d d i t i o n t o t h e

f o r m a t i o n o f B O 4 t e t r a h e d r a , i o n i c a t t r a c t i o n b e t w e e n a d j a c e n t s h e e t s o f B O 3 tr i-

a n g le s a f t e r B - O b o n d s a r e b r o k e n p r o b a b l y a l s o c o n t r i b u t e s t o t hi s in c r ea s e .


10/11

44

A. Makishima, J.D. Mackenzie, Young s modulus of glass

I 0 0 0

~C

8 0 0

~

6 0 0

'o

o

o ~ 4 0 0

c

D

2 0 0

N a 20-O aO - B ,203 -S i 02

/°

N a 2 0 - B 2 0~ S 02

PbO-B203

,o, x

I I I I

0 2 0 4 0 6 0 8 0 I 0 0

B 2 0 3 m o l e °./o

Fig. 4. R elationship between observed Young's modulus and B 203 content of borate glasses

[9(a), 10].

3 . 3 . A p p l i c a t i o n s o f t h e p r e s e n t w o r k

T h e r e s u lt s o f t h e p r e s e n t s t u d y c a n b e u ti li z e d t o p r e d i c t c h e m i c a l c o m p o s i t i o n s

which wi l l g ive g l asses wi th the h ighes t poss ib l e Young ' s modulus . In th i s respec t ,

f ro m eq . (11 ) , t he ox i des w i t h h i gh d i s soc i a t i on ene rgy pe r un i t vo l um e a re ' c and i -

da t e s ' f o r p repa r i ng a g l a s s w i t h t he h i ghes t Y oung ' s m odu l us . T hese ox i des a r e B eO ,

A 120 3 , Z rO 2 , C aO , M gO , w hose va l ues o f G a re h ighe r t han t ha t o f s il ic a .

T h e h i g h e st Y o u n g ' s m o d u l u s p r e d i c t e d b y W i ll ia m s e t a l. [ 3 ] f o r a m i x t u r e o f

40 % 0 Z rO 2 and 6 0 ° / 0 o B eO i s abo u t 19 .3 X 102 kb a r . H ow ev e r , i t is un l i ke l y t ha t

such a m i x t u re w i ll f o r m g la ss ea s il y . W e cons i de r t ha t t he h i ghes t va l ues w ou l d be

c l o se r t o 16 × 10 2 kb a r by a s sum i ng tha t t he pack i ng dens i t y is ab ou t 65% and

Y.iGiXi

= 2 7 . 5 k c a l / c m 3 , w h o s e m a i n o x i d e c o n s t i tu e n t s a r e B e O , A I 2 0 3 a n d Z r O 2 .

I t i s app a re n t t ha t t he ex ac t p red i c t i on o f Y o ung ' s m odu l us o f g la sses is a ve ry

d i f f i cu l t p rob l em as t he i r s t ruc t u re s a r e no t quan t i t a t i ve l y de f i nab l e a s fo r c ry s -

tal l ine sol ids.

I t is kno w n t ha t t he ha rdnes s o f g la ss is c lo se l y r e l a t ed t o t he ¥ o un g ' s m o du l us .

T he p re sen t l y deve l op ed eq . (11 ) is t hus app l i cab l e to t he p red i c t i on o f ha rdness


11/11

A. Makishima, J.D. Mackenzie, Young s modulus of glass 45

and to the preparation of hard glass. A large number of hard glasses with Vicker's

hardness nu mber as high as 1000 kg/mm 2 have been rece ntly prepared by this

approach [12].

Acknowledgements

This work has been support ed by the Directorate of Chemical Sciences, AFOSR,

under Grant No. 70-I 856 for which we are most grateful.

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

paper and to W. Capps for providing us with unpu bli she d density data.

References

[ 1 ] W. Capps and D.H. Blackburn, The Development of Glass Fibers having High Young's

Moduli of Elasticity, Nat'l. Bur. Std. Rept. 5188 (1957).

[2] C.J. Phillips, Glass Tech., 5 (1964) 216.

[3] M.L. Williamsand G.E. Scott, Glass Tech., 11 (1970) 76.

[4] R.J. Charles, in: Progress in Ceramic Science, fol. 1, ed. J.E. Burke (Pergamon, Elmsford,

N.Y., 1961) pp. 1-38.

[5] K.L. Lowenstein, Phys. Chem. Glasses, 2 (1961) 69.

[6] (a)

J.J. Gilman, Micromechanics of Flow in Solids, (McGraw-Hill,New York, 1969)

pp. 29-42;

(b)

J.J. Gilman, Mechanical Behavior of Ionic Crystals, in: Progress in Ceramic Science,

Vol. 1, ed. J.E. Burke (Pergamon, Elmsford, N.Y., 1961);

(C) J.J. Gilman, Mechanical Behavior of Crystalline Solids, Proceedings of American

Ceramic Society Symposium, Nat'l. Bur. Std. Monograph 59, (1963) pp. 79-102.

[7] K.H. Sun, J. Amer. Ceram. Soc., 30 (1947) 277.

[8] L. Pauling, Nature of Chemical Bond and Structure of Molecules and Crystals, 2nd ed.

(Cornell University Press, Ithaca, N.Y., 1940).

[9] (a) S. Spinner, J. Amer. Ceram. Soc., 37 (1954) 229;

(b) T.J. Sokolowski and M.H. Manghneni, J. Amer. Ceram. Soc., 52 (1971) 539;

(c) S.D. Brown and G.Y. Onoda, Jr., High Modulus Glasses Based on Ceramic Oxides,

Final Report for Bureau of Naval Weapons Department of the Navy (1966);

(d) M.H. Manghneni, J. Amer. Ceram. Soc., 55 (1972) 360;

(e) M.B. Field and R.W. Tucker, J. Amer. Ceram. Soc., 54 (1971) 309.

[10] R.R. Shaw and D.R. Uhlmann, J. Non-Crystalline Solids, 5 (1971) 237.

[11 ] P.J. Bray and J.G. O'Keefe, Phys. Chem. Glasses, 4 (1963) 37.

[12] K. Park, A. Makishima and J.D. Mackenzie, to be published.

Documents

Direct Calculation of Young's Modulus of Glass