Thermodynamic behaviour of undercooled melts

Bull. Mater. Sci., Vol. 19, No. 2, April 1996, pp. 357-371. ,~, Printed in India.

Thermodynamic behaviour of undercooled melts

R K MISHRA and K S DUBEY Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India

MS received 24 November 1995; revised 19 December 1995

Abstract. The Gibbs free energy difference (AG) between the undercooled liquid and the equilibrium solid phases has been studied for the various kinds of glass forming melts such as metallic, molecular and oxides melts using the hole theory of liquids and an excellent agreement is found between calculated and experimental values of AG. The study is made for non-glass forming melts also. The temperature dependence of enthalpy difference (AH) and entropy difference (AS) between the two phases, liquid and solid, has also been studied. The Kauzmann temperature (T 0) has been estimated using the expression for AS and a linear relation is found between the reduced glass transition temperature (T / T ) and (To/T). The residual entropy (AS~} has been estimated for glass forming melts anc] an attempt is made to correlate AS a, T, T o, and T which play a very important role in the study of glass forming melts.

Keywords. Undercooled melts; Gibbs free energy; Kauzmann temperature; metallic glasses; residual entropy.

I. Introduction

The production of metallic glasses by rapid solidification techniques has recently received sufficient attention and resulted in a revival of interests in nucleation and growth kinetics in liquid alloys, The Gibbs free energy difference (AG) between the undercooled liquid and the equilibrium solid phases is found to be an important parameter in the classical theory of nucleation and growth processes while the entropy difference (AS) between the liquid and solid phases plays an important role in the viscosity of these melts. The Gibbs free energy difference (AG) can be estimated by using the thermodynamic relations from a knowledge of the experimentally measured values of specific heat difference (ACp) between the undercooled melt and equilibrium solid phases. But the experimental measurements of the specific heat of the glass forming melts in their super-cooled states can be carried out only in a restricted temperature range below the melting temperature (Tm) due to their strong tendency to crystalliza- tion which prevents any direct measurements of the thermodynamic quantities. It can at best be measured near the melting temperature and at the glass transition temperature (T~). Often, the behaviour of the thermodynamic parameters from Tg to T., is obtained by extrapolating the high temperature data to low temperature region.

Due to experimental problems, several investigators (Turnbull 1950; Hoffman 1958; Jones and Chadwick 1971; Thompson and Spaepen 1979) have tried to suggest appropriate analytical expressions for AG in terms of experimentally measured parameters such as entropy of fusion (ASm), T m, etc. Most of these expressions are inadequate for describing the temperature dependence of AG over a large degree of undercoolings.

Recently an expression for AG has been obtained by Dubey and Ramachandrarao (1984) based on the hole theory of liquids. Following the earlier work of Dubey and his coworkers (Dubey and Ramachandrarao 1984: Pandey et al 1987), the aim of the

357

358 R K Mishra and K S Dubey

present work is to study AG for various kinds of glass forming melts such as metallic, molecular and oxide glass formers. AG has been estimated for AuvTGe18.6Si9. 4, Ausl.4Sils.6, Mgas.sCu14.5, Mgal.6Ga18. 4, glycerol, ethanol, 2-methyl pentane, B 20 3 , SiO 2 and GeO 2 in the temperature range Tg to T m and an excellent agreement is obtained between calculated and experimental values of AG. The study is also made for three samples of non-glass forming melts, Pb, Ga and BissPb45, in the temperature range 0"5 T m to T m. The expressions for the entropy difference (AS) and enthalpy difference (AH) between the liquid and solid phases have been derived using the hole theory of liquids and the applicability of the expression obtained has been tested by estimating AS and AH for one sample from each kind of glass forming melts Mg85.sCU14. 5 (metallic), glycerol (molecular), and SiO 2 (oxide) and one sample of non-glass forming melt, Pb. A very good agreement is found between calculated and experimental data.

The expression derived for AS has been further used to obtain an expression for the Kauzmann temperature T O (which is also known as the ideal glass transition temperature) as well as for the residual entropy (ASR) of glass forming melts and T O and AS R are calculated for the above stated glass forming melts. Attempt has been made to correlate T O and Tg and a linear relation has been obtained between T J T m and To/T . for each kind of glass forming melts. The role of the characteristic temperatures T o, T, and T m in the estimation of the 'frozen-in' entropy (ASR) has been studied and a linear relation is found between ASR/AS m and (Tg - To)/T m. Attempt has also been made to study the variation of the residual entropy with the ratio (ACp"/ASm) of the specific heat and entropy difference between two liquid and solid phases, at T = T m. The glass forming ability of material has been discussed in terms of the residual entropy of melts.

2. Free energy, enthalpy and entropy of undercooled liquids

According to the hole theory, a liquid state is described as a quasi-crystalline lattice with a considerable fraction of vacant sites of holes and the hole concentration is usually derived by minimizing the change in the free energy associated with the introduction of holes into a lattice. Following Frenkel (1955) as well as Hirai and Eyring (1958, 1959), the change in the free energy (AG) due to introduction of N h hole into a lattice having N~ atoms or molecules at a temperature T can be expressed as

N h nN~ 7 AG = Nh(~la -{- PVh) + K a T N h In Nh + nN~ + Nh In Nh --+ -rtN~ J[ -- TASh'

(1)

here gh is the energy required for the formation of a hole of volume v h, p the external pressure, K a represents the Boltzmann constant, n -- v~/v, is a measure of the relative volume of a hole and an atom, v~ the hard core volume per atom or molecule and AS h the entropy change associated with the formation of a hole.

Following Flory (1953), Sanchez (1974) and Dubey and Ramachandrarao (1984) and minimizing the AG, an expression for N h can be obtained as

N h = nS~(g/(1 - g)), (2)

Thermodynamic behaviour of undercooled melts 359

with A = 1 - l/n.

The temperature dependence of ACp can be derived by realizing ACp = d/dt(AH). According to Flory (1953) and Sanchez (1974), the enthalpy change associated with the formation of hole is given by

AH=(I --g)ShNh,

and can be used to obtain the temperature dependence of the specific heat difference between the liquid and solid phases. Realising ACp = d/dt(AH), and following Dubey and Ramachandrarao (1984) one can have

E h )2 ACp= nR ~-~ g, (4)

here R is the gas constant and E h the hole formation energy per mole. It is usually found that ACp increases with increase in the undercooling below T m.

The process of increase in ACp cannot go on indefinitely and as per Kauzmann paradox, ACp should attain a maximum value at some undercooled temperature to avoid the entropy catastrophe, The temperature at which ACp becomes maximum can be assumed to be the same at which AS becomes zero, i.e. the ideal glass transition temperature T o. As a result, one gets

Eh/RT o = 2. (5)

Similar relation has also been obtained by Sanchez (1974) in connection with the viscous behaviour of polymeric melts. Consequently, (4) reduces to

ACp = ACp"(Tm/T)2 e - 26A'1",,1" (6)

with 6 = To/T m is the reduced ideal glass transition temperature. The expression for ACp stated in (6) can be used to derive an expression for AG with

the aid of basic thermodynamic relations

AG = AH - TAS, (7) ;;m AH = AH m - ACpdT, (8)

AS = AS m - (ACp/T)dr. (9)

Appropriate substitution of (6) and simplification of (7) yield an expression for AG as

AG = A S m A T - ACp~[26AT- T(I - e 26AT~T)], 482

(10)

under the approximations,

3--X 1- -e 2-' = 2 x - - ( l la)

3 + 2 x '

360

and

R K Mishra and K S Dubey

3 + x e 2x- 1 = 2 x 3 _ 2 x , ( l lb)

the expression for AG stated in (10) reduces to

AT z 2 ' (12) AG=ASmAT-ACpm + 5(6AT~T))

which can further be simplified as

ACm AT2 f l AG=ASmAT- P - ' ~ - t - 5 6 ~ T ) " (13) 2

The expression stated in (13) requires the knowledge of A Cp m, AS m and T m which can be measured easily. The parameter 6 can also be estimated with the aid of ACp" and AS m which has been discussed in the next section.

The use of (6) and (9) yields an expression for AS as

ACp m AS = AS m - 46~-T- [(1 + 26) - (1 + 26(Tm/T))e-2~Ar/r]. (14)

In view of the approximation stated in (11), expression for AS takes the form

AS=ASm ACpmAT(Tm2T 2 + T) II-(2/3)6(AT/TJ(TJ(Tml + t2/3)6(AT/T) + T))], (15)

which can be further approximated as

AS=ASm ACpmAT(Tm"~- T)II 2~AT(2TmTt- T)I 2 T 2 - -5 - T - \ T-~+7 ~ " (16)

The expression for AH can also be derived with the aid of(6) and (8) and takes the form

ACp'nTm'" e - 2°aT/T) (17) AH = AH m ~ (1 - .

Use of approximation stated in (11) yields to a more simplified form

Thus, AS and AH can be estimated with the aid of AS m, ACp m and T m which can be measured easily.

3. The ideal glass transition temperature T o and the residual entropy AS s

The problem associated with a purely kinetic definition of the glass transition was pointed out by Kauzmann (1948). One of the problems is AS becoming negative at some

Thermodynamic behaviour of undercooled melts 361

temperature below Tg. He further argued that a liquid loses its entropy at a faster rate than the corresponding equilibrium solid. Consequently one can assume an iso- entropic temperature (To) at which both liquid and solid phases have equal entropy. Gibbs and DiMarzio (1958) as well as Adam and Gibbs (1965) have pleaded that such a transition should be of the second order while Cohen and Grest (1979) reported in favour of the first order. Recalling (14) and setting to zero at T = T o, one can have an expression for the estimation of T O in terms of ACp m, AS m and T m of the form

ACp" -~1] = 0. (19) AS m - 2 ~ [ ( 1 + 26 ) - 3e 2(1

The solution of the transcendental equation can be achieved by iterative procedure. However an approximate and more simplified form of the expression for estimating 6 can be obtained under the approximation stated in (11) as

I 5ACpm~62 4ACp" ACp m 463 -- 10 + AS m J + ~ 6 q AS~- - 0, (20)

which shows that the value of 6 is entirely governed by the ratio ACpm/ASm . It may be noted that generally 63 is found to be much smaller compared to 62 and an approximate value of the ideal glass transition temperature T O is given by

2s + (9S 2 + 10S) 1/2 (21) T° = Tm 5(2 + S)

where ACp m

S : - - ASm

According to the statistico-mechanical quasi-lattice theory of Gibbs and DiMarzio (1958), the configurational entropy becomes zero at the iso-entropic temperature T o. The apparent entropy difference between the hypothetical glass and crystal at T - 0 K is the same as the configurational entropy of the liquid frozen-in at T~. In other words, it can also be understood that some amount of entropy is blocked in the glassy configuration at Tg and is usually referred to as the residual entropy or 'frozen-in' entropy. Thus the residual entropy is given by

where

Use of(1 I)

which can

ASR=AS ( a t T = 0 K ) = A S ( T g )

Act[f462 Tin) -- c ~,l -t- 2(~ 1, je--2'iAr'lT'-- 3e--2tl {22)

A T , = ( T m - T,).

yields a simplified form of AS R as

AS, = ACp" [1.5 - 2(TolTg)+ 0"5(TolT,)Z]e - 2~1- a~. (23)

also be stated as

ACpm T~-2-1 7 6 ( 2 T g r - 3 T ~ r 2 - 1 ) (24) AS s = A S m 1 A S m - -


as a function of ACp" and AS m. Here Tg r = Tg/T m is the reduced glass transition temperature.

Expressions stated in (21-24) show that the residual entropy is mainly governed by the characteristic temperatures Tg, T m and T o.

4. Results and discussions

The experimental values of AG have been calculated with the help of experimentally measured values of ACp and using (7). Generally, experimental data of ACp is represented by the equation

ACp= A + BT + CTZ + D/T + E/T 2, (25)

where A, B, C, D and E are constants. The material constants are reported in tables la and b. To test the expression for AG reported in (13), AG has been calculated for various types of glass forming melts in the temperature range T m to Tg, which includes AuvvGel3.6Si9.4, Ausx.4Six8.6, Mgss.sCu14. 5 and Mgsl.6Ga18. 4 (metallic glass formers), glycerol, 2-methyl pentane and ethanol (molecular glass formers) and B203, GeO 2 and SiO 2 (oxide glass formers) and results obtained are illustrated in figures 1-3 for metallic, molecular and oxide glass forming melts respectively. To see the applicability of the expression reported in (13) for non-glass forming melts, AG is also calculated for Ga, Pb and Bi55 Pb45 in the temperature range T m to Tm/2 and results obtained are shown in figure 4. An excellent agreement between calculated and experimental values of AG can be seen in figures 1-4 for the glass forming as well as for non-glass forming melts.

The temperature dependence of the thermodynamic parameters AS and AH has been studied for one sample of each group of glass forming melts; MgCu (metallic), glyceral (molecular), SiO 2 (oxide) in the temperature range T m to Tg and one sample of non-glass forming melt Pb in the temperature range T m to Tin~2 and are illustrated in figures 5-8 respectively. From figures 5-8, it can be seen that the agreement between calculated and experimental values of AH is very good for glass forming as well as for non-glass forming melts. With the help of these figures (figures 5-8), it can also be realized that the calculated values of AS are quite close to the experimental values for materials under study except Pb. The agreement between the calculated and experimental values of AS is not very satisfactory for Pb at the large undercoolings. As a result of it, one can say that expressions reported in (16) and (18) are capable to explain the temperature dependence of the thermodynamic parameters AS and AH respectively. To see the goodness of the expression reported in (13) for AG, a comparative study is made in table 2 for Mgss.sCu14. s with the results obtained using the expressions obtained by earlier workers (Turnbull 1950; Hoffman 1958; Thompson and Spaepen 1979) as

AG = ASmA T, (26)

T AG = ASmA TT~ m, (27)

2T AG = ASmA T)T + T)" (28)

Tab

le l

a.

Th

e m

ater

ial

par

amet

ers

use

d i

n th

e st

ud

y of

th

e th

erm

od

yn

am

ic

par

amet

ers

AG

, A

H,

AS

, T

O a

nd

AS

R.

Mat

eria

l

Pa

ram

eter

A

usl

.4S

ila.

6 A

uv~G

ely6

Si9

4 M

gas.

sCu

l,~

.5

Mg

al.

6G

ala

.4

Bis

s P

b4s

G

a

Pb

,...]

a (J

/mo

l/K

) 26

-618

6 32

-526

4 15

.17

17.7

5 0.

22

4.30

28

8.61

49

b (J

/mol

/K z

) 103

-

53,7

64

- 63

.082

2 -

9,88

-

12-8

3 -

16'2

343

- 12

,162

9

c (J

/mo

l/K

3)

105

3.29

49

3.41

41

- -

1.00

82

--

0'20

53

d (J

/mo

l)lO

2

__

1-

3004

--

--

11

'190

1 --

e (J

K/m

ol)

10

-4

--

2"51

88

- --

--

AH

m (

J/m

ol)

9445

10

627

7838

7

12

0

4178

55

89

48

02

AS

m (

J/m

ol/

K)

15.4

8 17

-03

10.3

4 10

-23

10.5

0 18

.45

7-99

AC

p m

(J/t

ool/

K)

7.75

6.

74

7-68

3.

94

0.22

2'

89

1.21

T m

(K

) 63

6 62

5 75

8 69

6 39

8 30

3 60

1

T

(K)

290

294

380

390

--

--

g:::

e~

t~


Table lb. The material parameters used in the study of the thermodynamic parameters AG, AH, AS, T O a n d ASR.

Material

2-methyl Parameter Glycerol Ethanol pentane B20 3 SiO 2 GeO 2

a(J/mol /K) 90-8646 51.5810 146-871 70-854 14-20 20-98 b(J/mol/K2)lO a 39"06 - 167.3 -798"5 -73.416 - 1.9 - 9 . 8 7 c (J/mol/K 3) l0 s . . . . . . d (J/tool) 10- 2 . . . . . . e (JK/mol) 10 -4 - - - - - - 141.12 392'1 177.7 AH m (J/mol) 18371 5015 6260 22261 9621 44517 AS,, (J/mol/K) 62.70 31.64 52-36 30.79 4.82 32'05 ACp m (J/mol/K) 79'42 25"06 51-41 20.47 11-39 8.19 T m (K) 293 158.5 119.55 723 1996 1389 Tg (K) 186 91 79.5 550 1443 823

~ 3

E

Au77Get3'6S~/ i18.6

Mg8t6 ~ct18.t,

14.. ~

• I 1 i I ~ i i i 100 200 300 400

AT(K)

Figure 1. The Gibbs free energy difference, AG, for metallic glass formers. Closed circles, triangles, open circles and squares represent experimental values of AG for Mg85.sCu~4.s, Mgsl.6Gals.4 , Ausl.4Sils.6 , Au77Ge 13.6Si9.4 respectively whereas lines are calculated values of the respective samples.

From table 2, it can be seen that the results obtained on the basis of expression stated in (13) are the best and values of AG derived on the basis of(13) are the closest to the experimental values even for large undercoolings. At T = Tg the error is only about 3"4% at the undercooling A T = 370K, while it is 36"6%, 31.5% and 8"8%, based on expressions proposed by Turnbull, Hoffman, and Thompson and Spaepen respectively.

3

I 0 120

Ethanol

I I I I I I I I I I 20 40 60 80 100

~T(K)

Figure 2. The Gibbs free energy difference, AG for molecular glass formers. Closed circles, open circles and squares represent experimental values of z~G for glycerol, 2-methyl pentane and ethanol respectively whereas lines are calculated values of the respective samples.

'I

12

o E

~° 8

Ge Oz

16-

~ SJOz _ Si Oz .-o ,r

I t I I I 0 100 200 400 500 300 600

&T(K)

6 c:~

¢...

o

Figure 3. The Gibbs free energy difference, AG, for oxide glass formers. Squares, closed circles and open circles represent experimental values of AG for SiO 2, GeO 2 and B203 respectively whereas lines are calculated values of the respective samples.

366 R KMishra and KS Dubey

3.2

2.4

E1.6

0.8

GQ

• P b

I I I

80 160 240 320 AT(K)

Figure 4. The Gibbs free energy difference, AG, for non-glass forming melts. Closed circles and lines represent experimental and calculated values of AG of the respective samples.

0

E

v

~4

I

0 400

Cu14. G

I , ~o , ,~o ~

100 200 300 At(K)

Figure 5. Variation of AG, AH and AS with undercooling A T for metallic melt Mg s s. s Cu ~ ,.5. Circles represent experimental data of respective parameters while lines are calculated values.

Thus, the express ion for AG s ta ted in (13) can be r e c o m m e n d e d in the use of nuc lea t ion and growth techniques. At the same time, it is in teres t ing to note tha t present

express ion requires ACp m, AS m and T m as input da t a which can be measured easily and more accura te ly and does not con ta in any ad jus tab le pa ramete r .

Thermodynamic behaciour of undereooled melts

20

367

16 -

--.<... "l-

~ s

I I I AG 4 1 ZO 6 0 $ 0 80 100 120

1 ¢ . ' I " - t I l I I I I I I I I 20 40 60 BO 100 120

6T(K)

Figure 6. Variation of AG, AH and AS with undercooling A T for molecular melt glycerol. Circles represent experimental values of respective parameters whereas lines are calculated values.

10

8

2

0 600

0 200 400 600 AG A T ( K ) ~ - - ' - ' - - ~

/ i I t 1 [ 200 400

~T(K)

Figure 7. Variation of AG, AH and AS with undercooling AT for oxide melt SiO 2. Circles represent experimental values of respective parameters whereas lines are calculated values.


~3

Pb

- 80 160 240 320

I I I I I I i 80 160 240

AT(K) 320

Figure 8. Variation ofAG, AH and AS with undercooling AT for non-glass forming melt Pb. Circles represent experimental data of respective parameters while lines are calculated values.

The ideal glass transition temperature T O is calculated using (19) for glass forming melts and values obtained are reported in table 3. From table 3, it can be seen that the value of T O is very close to the extrapolated value of T O based on the experimental data. It can also be seen that the material with larger value of ACpm/ASm exhibits a larger value of 6 = To/T m. During the analysis, it is also found that T O is mainly controlled by the ratio ACpm/ASm . An attempt has been made to relate Tg and T O and a linear relation

T g / T m = a ( T o / T m ) q- b, (29)

is found between the reduced glass transition temperature and the reduced ideal glass transition temperature for each kind of glass forming melts. The values of constants a and b are reported in table 4 for metallic, molecular and oxide melts. A similar relation is also reported by Dubey and Ramachandrarao (1992) and Adam and Gibbs (1965) for organic materials.

The residual entropy (ASa) of glass forming melts has been estimated and results are stated in table 3. AS a is also evaluated using experimental data of ACp. It is found that calculated values of AS a is close to that based on experimental data. Attempt has also been made to seek a relation between ASa and characteristic temperatures T o, Tg and T m. A linear relation has been obtained between A S s ~ A S m and (Tg - To)/T m of the form

A S a / A S m = m (Tg - T°) + n, (30) 7" m

where m and n are constants and are reported in table 4. These constants are found to be different for different kinds of glass forming melts. From table 3, it can be realized

Tab

le 2

. F

ree

ener

gy d

iffe

renc

e A

G b

etw

een

the

liqu

id a

nd

equ

ilib

rium

sol

id p

hase

s of

Mga

s.sC

u~4

5 ev

alua

ted

from

vari

ous

expr

essi

ons a

nd c

ompa

red

wit

h th

at c

alcu

late

d us

ing

(13)

. The

exp

erim

enta

l val

ue w

as e

valu

ated

usi

ng e

xper

imen

-

tal

dat

a of

AC

p.

AG

(J/to

ol)

Tur

nbul

l (1

950)

H

offm

an (

19

58

) T

ho

mp

son

and

Spa

epen

(19

79)

T(K

) A

T(K

) E

quat

ion

(26

) E

quat

ion

(277

E

quat

ion

(28)

E

quat

ion

~13

) E

xper

imen

tal

750

8 83

82

83

82

82

70

0 58

60

0 55

4 57

6 58

2 58

2 65

0 10

8 11

17

958

1031

10

51

1052

60

0 15

8 16

34

1293

14

44

1484

14

88

550

208

2151

15

61

1809

18

76

1885

50

0 25

8 26

68

1760

21

21

2217

22

39

450

308

3185

18

91

2373

24

99

2541

40

0 35

8 37

02

1953

25

57

2743

27

83

380

378

3909

19

59

2610

27

65

2861

t%

Tab

le 3

. T

he v

alue

T O

and

AS~

for

var

ious

gla

ss f

orm

ing

sam

ples

use

d in

the

pre

sent

ana

lysi

s.

as~

~,

as~

'~

T,

- L

Tm

(K )

T,(

K)

T~

a~(K

) T

~xP(

K)

6 (J

/mol

/K)

(J/m

ol/K

) A

SCaP

/S,.

AC

pm/A

Sm

Tm

Mat

eria

l

Gly

cero

l 29

3 18

6 15

5.4

13

7-4

0.

4689

25

-59

19-6

6 0.

4081

1

.26

70

6

-16

58

E

than

ol

158.

5 91

66

.9

63

-1

0.3

98

1

14.2

3 13

.22

0.44

97

0.7

92

6

0-1

76

0

2-m

ethy

l pe

ntan

e 11

9-55

79

.5

56.2

6

0.8

2

0-5

08

7

24.4

2 26

.94

0-46

64

0.9

81

8

0-1

56

2

B20

3 72

3 55

0 27

7-5

336

0-46

47

23.1

3 24

-21

0-75

12

0.6

64

8

0.2

95

9

SiO

2 19

96

1443

13

50

1358

0.

6803

0.

8144

0.

8172

0.

1689

2-

363

0.04

26

GeO

2 13

89

823

317

29

0-5

0

.20

91

25

-81

25.6

4 0-

8053

0

.25

55

0

-38

33

A

ual.

4Sil

a. 6

63

6 29

0 21

0"5

202

0-31

76

6.33

5.

77

0-40

88

0.50

06

0.13

84

Auv

TG

el 3

.65i

9.4

625

294

181.

5 1

99

.4

0-3

19

0

7-84

8-

66

0-46

01

0.3

95

7

0.1

51

4

Mga

s.sC

u14

s 75

8 38

0 30

0 28

1 0.

3707

3.

60

2.76

0.

3480

0

.74

27

0.

1306

M

gsl

.6G

ata

~ 69

6 39

0 19

9-5

25

1.5

0

-36

13

5.

55

6-84

0-

5426

0-

3851

0.

1990


Table 4. The values of constants a, b, m, and n.

Kinds of materials a b m n

Metallic 1.27551 006056 2.5550 0-04423 Molecular 0 . 8 2 5 2 1 0.24618 - 0.79223 0.57291 Oxide 0-28842 0'56185 1.96431 0.10252

that material having smaller value of (Tg - To)/T m shows a lower value of ASR/AS m

which can be explained as follows. The material having lower value of AS R requires trapping of less amount of entropy to form a glass and such material can exhibit T o near Tg i.e. material shows less value of (Tg - To)IT m. It can also be seen that the material with lower value of ASR/AS m shows a larger value of ACp" /AS m. According to earlier report of Dubey and Ramachandrarao (1984-85), the material having large value of ACp" /AS m is a good glass former. In view of these findings, one can say that the material with lower value of ASR/AS m exhibits a large value of 6 and ACpm/ASm and is a good glass former. Thus, it can be said that 6, ACp" /AS m and ASR/AS m are dependent on each other and can be used to measure the glass forming ability of materials.

5. Conclusion

The expression for AG reported in (13) is capable to estimate AG of all kinds of materials correctly at large undercooling also and expressions reported in (16) and (18) for AS and AH respectively can be used to study the temperature variation of these parameters. The value of the Kauzmann temperature T o is found to be close to the value extrapolated on the basis of experimental data. A linear relation is found between Tg/T m and To/T m for glass forming melts. Both, reduced ideal glass transition temperature 6( -- To/T m) and reduced residual entropy (ASR/ASm) are mainly controlled by the parameter ACpm/ASm and T m. A linear relation is also found between ASR/AS m and (Tg-To) /T m. The material having lower value of ASR/AS m can form glass more easily. AS R, T o and ACp" /AS m are dependent on each other and the glass forming ability of materials can be studied in the form of these parameters.

Acknowledgement

The authors wish to express their thanks to Prof. P Ramachandrarao, NML, Jamshed- pur and Prof. S Lele, Department of Metallurgical Engineering, BHU, Varanasi for their interests in the present work. Part of the work was done under the financial support by CSIR, New Delhi and authors are thankful for that.

References

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Thermodynamic behaviour o f undercooled melts 371

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Thermodynamic behaviour of undercooled melts