A kinetic treatment of glass formation

JOURNALOF NON-CRYSTALLINESOLIDS7 (1972) 337-348 © North-Holland Publishing Co.

A K I N E T I C T R E A T M E N T OF GLASS F O R M A T I O N D. R. UHLMANN Department of Metallurgy and Materials Science, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.

Received 4 November 1971 A kinetic treatment of glass formation is presented. This treatment is based on the construction of time-temperature-transformation curves corresponding to some barely detectable degree of crystallinity. From such curves, the minimum cooling rates required to form glasses of various materials are estimated. The most important factors determining the glass-forming abilities of different materials are suggested to be the magnitude of the viscosity at the melting point and the rate of increase in viscosity with falling temperature below the melting point. 1. Introduction In the past decades there have been several treatments of the conditions for glass formation based on considerations of crystallization kinetics (refs. 1-3, e.g.). The first of these treeatments led to the conclusion that a pure liquid in bulk form, free of nucleating heterogeneities, would not crystallize if the kinetic barrier to either nucleation or crystal growth exceeds 30 RTE (where R is the gas constant and TE is the melting point.) Subsequent discussion 4, 5) has directed attention to the possible importance of transients on the frequency of nucleation. It was noted that the times required to build up the steady-state concentrations of subcritical embryos can be quite long (estimated in the range 106-107 sec) as the glass transition is approached. These times can thus be longer than the usual experimental times, and can effectively preclude the reliable determination of nucleation kinetics in many systems. A parallel series of developments based on kinetic and statistical mechanical 6, 7) treatments of transport and molecular rearrangement in liquids have suggested that the existence of a glass transition may be a universal feature of liquid behavior - provided that the liquid can be undercooled to a sufficient ly low temperature without the occurrence of crystallization. More recently, Turnbull8) has readdressed his attention to the conditions under which glasses may be formed. He noted that there are at least some 337

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D.R.UHLMANN

glass formers in every category of material based on bond type (covalent, ionic, metallic, van der Waals, and hydrogen). The cooling rate, density of nuclei and various material properties were suggested as significant factors which affect the tendency of different liquids to form glasses. This approach leads naturally to posing the question not w h e t h e r a material will form an amorphous solid when cooled in bulk form from the liquid state, but rather h o w f a s t must a given liquid be cooled in order that detectable crystallization be avoided. In turn, the estimation of a necessary cooling rate reduces to two questions: (1) how small a volume fraction of crystals embedded in a glassy matrix can be detected and identified; and (2) how can the volume fraction of crystals be related to the kinetic constants describing the nucleation and growth processes, and how can these kinetic constants in turn be related to readily-measurable parameters? In answering the first of these questions, we shall concern ourselves with crystals which are distributed randomly through the bulk of the liquid, and will identify a volume fraction of 10 - 6 a s a just-detectable concentration of crystals. In answering the second question, we shall adopt the formal theory of transformation kinetics developed in the late 1930's by Johnson and Mehl 9) and Avrami 10-12). 2. Transformation kinetics

In the present paper we shall be concerned with single-component materials or congruently-melting compounds, and will assume that the rate of crystal growth and the nucleation frequency are constant with time. For such a case, the volume fraction, X, crystallized in a time, t, may for small Xbe expressed:

X ~ i3~Lu3t'.

(1)

Here Iv is the nucleation frequency per unit volume and u is the rate of advaoce of the crystal-liquid interfaces per unit area of the interfaces. In identifying/~ as the steady-state rate of homogeneous nucleation, we shall neglect heterogeneous nucleation events - such as at external surfaces and will be concerned with minimum cooling rates for glass formation. Clearly, a glass cannot be formed if observable amounts of crystals form in the interiors of samples. We shall also neglect the effect of transients during which the steady-state concentrations of subcritical embryos are built up by a series of bimolecular reactions. The characteristic transient time for the nucleation process is given to order-of-magnitude accuracy as 18,14). z ~

(n*) Nsv

= Kq,

(2)

339

KINETIC TREATMENT OF GLASS FORMATION

where n* is the number of atoms in the nucleus of critical size, Ns is the number of atoms on its surface, v is the frequency of transport at the nucleusliquid interface, and r/is the melt viscosity. Neglect of transients in the present analysis is justified whenever the time required to establish the steady-state nucleation rate is small relative to the total transformation time. Since latter quantity is also proportional to the viscosity, transient effects will only be important here when the barrier to nucleation is unusually large or when the frequency factor for transport at the macroscopic crystal-liquid interface (in crystal growth) is much larger than v. While either of these conditions may maintain for particular systems, they are not expected to maintain with any generality. The cooling rate required to avoid a given volume fraction crystallized may be estimated from eq. (1) by the construction of so-called T-T-T (time-temperature-transformation) curves, an example of which is shown in fig. 1 for two different volume fractions crystallized. In constructing these curves, a particular fraction crystallized is selected, the time required for that volume fraction to form at a given temperature is calculated and the calculations is repeated for other temperatures (and possibly other fractions crystallized). The nose in a T - T - T curve, corresponding to the least time for the given volume fraction to crystallize, results from a competition between the driving force for crystallization, which increases with decreasing temperature, and the atomic mobility, which decreases with decreasing temperature. The trans-

2o],

40

6o

eo

I I0 t

I

I

I

I

I

I0

I01

I0 z

I0 ~

104

t

TIME ( S E C )

Fig. 1. Time-temperature-transformation curves for salol corresponding to volume fractions crystallizedof (A) 10-8 and (B) 10-s.

340

D.R.UHLMANN

formation times, ti, are relatively long in the vicinity of the melting point as well as at low temperatures; and for purposes of the present paper, we shall approximate the cooling rate required to avoid a given fraction crystallized by the relation:

ci

e

"gN

(3)

where ATN=TE-TN; TN=temperature at the nose of the T - T - T curve; ZN= time at the nose of the T - T - T curve; and T~ is the melting point. From the form of eq. (1), as well as from the curves shown in fig. 1 which were calculated therefrom, it is apparent that the cooling rate required for glass formation is rather insensitive to the assumed volume fraction crystallized, since the time at any temperature on the T - T - T curve varies only as the one-fourth power of X. An alternative estimate of the glass-forming characteristics of materials may be obtained by considering the thickness of sample which can be obtained as an amorphous solid. Again using the criterion of a volume fraction crystallized less than 10-6, and neglecting problems associated with heat transfer at the external surfaces of the sample, the thickness, Yc, of sample which can be formed without detectable crystallization should be of the order: y¢ ~ (DTHZN) i(4) where DTa is the thermal diffusivity of the sample. In estimating the critical conditions for forming a glass of a given material, one can in principle employ measured values of the kinetic factors in calculating the T - T - T curves. In practice, however, information on the temperature dependence of the nucleation frequency is seldom available; and in only a portion of the cases of interest are adequate data available on the variation of the growth rate with temperature. In nearly all cases, therefore, it will be necessary to estimate the nucleation frequency form the standard model for homogeneous nucleation; and in some instances it will prove convenient to estimate the growth rate from standard theoretical models as well. For describing nucleation in glass-forming systems, the nucleation frequency may conveniently be expressed 4): - 1.024

Iv ~ NOv exp TraAT2 .

(5)

Here N ° is the number of single molecules per unit volume; Tr= T/TE; AT is the undercooling (AT= TE--T); and ATr=AT/T~.

KINETIC TREATMENT OF GLASS FORMATION

341

In obtaining this relation, it was assumed that the free energy of forming the critical nucleus is 50 k T at ATr=0.2 (in consonance with experimental results on a wide variety of materials) and that the motivating potential for crystallization, AG, could be related to the heat of fusion, AHf, by the relation AG = AHf ATrT r. The rate of advance of a crystal-liquid interface, per unit area of the interface, may be expressed 4): u = fvga o 1 - e x p

RT

JJ"

(6)

H e r e f i s the fraction of sites on the interface where atoms may preferentially be added and removed; Vgis the kinetic coefficient for transport at the crystalliquid interface; AHfM is the heat of fusion per gram atom and a0 is a molecular diameter. For materials with small entropies of fusion (AHfra/T E < 2R), the interface is expected 14,15) to be rough on an atomic scale a n d f s h o u i d be of the order unity and should not vary significantly with undercooling. For materials with large entropies of fusion (AHfM/T E > 4R), the interface is expected to be smooth on an atomic scale and growth should take place at steps provided by screw dislocations or two-dimensional nuclei formed on the interface. In the former case, which may be anticipated for imperfect crystals, f may be expressed: f ~ 0.2ATr.

(7)

In using eqs. (5) and (6) to evaluate the nucleation frequency and growth rate, it is customary to take v and vg as inversely related to the viscosity, ~/. For purposes of the present calculations, we shall take this proportionality coefficient as the Stokes-Einstein coefficient: v = v~ = b / ~ ,

(8)

where b = kT/3na3o.

While the particular value assumed for b may be open to some question, particularly in some of the important glass-forming systems, it is thought to provide a useful approximation in many instances, and should be quite accurate in the case of simple ionic and organic glass-forming liquids. 3. Application to particular systems In applying this treatment to particular systems, we shall consider two classic network oxides, SiO 2 and GeO2, as well as a relatively simple organic

342

D.R. UHLMANN

liquid, phenyl salicylate (salol). We shall also consider water and a metallic system in order to get some feeling for the cooling rates that might be required for glass formation in such materials. 3.1. SiO2 In this case, we shall use the viscosity data of Fontana and Plummer 16) together with the following values for the various parameters: TE= 1996 °K; a0=2.5 A; N ° = 2 x l 0 2 2 molecules c m - 3 ; AHfM=2415 cal (g-atom)-~; D T H = 1 0 - 1 c m 2 s e c - 1 ; and f = l . Using these values, the T - T - T curve ( X = 10 - 6 ) shown in fig. 2 was calculated for the formation of cristobalite. 1700

I

I

I

I

I

1

I

1 6 5 0 --

TO

200

x I 0 s a t T = 1 6 9 6 °C

1600

1550

1500

1450

--

1400 --

0

I

I

I

I

I

[

2

4

6

8

I0

12

I 14 x I06 SEC

Fig. 2. Time-temperature-transformation curve for SiOz corresponding to a volume fraction crystallized of 10-e.

From this, the critical cooling rate needed to form a glass [(dT/dt)c] is estimated as about 2 x 10 -4 °K sec -1, and the thickness of sample which can be formed as a glass [-yc] as about 400 cm. Using measured values of the growth rate 17) results in an estimated (dT/dt)¢ which is larger by about a factor of 5 and a y, value smaller by about a factor of 2. 3.2. GeO2 Here we shall again use the viscosity data of Fontana and Plummer, to-

KINETIC

TREATMENT

343

OF GLASS FORMATION

gether with: TE=1389 °K; ao=2.5 ,&; N ° = 2 . 1 X 1 0 2 2 molecules cm -3, AHfM= 3500 cal (g-atom)- ~; Da-H= l 0- 2 cm 2 sec- ~; and f = 1. Using these values, the appropriate T - T - T curve (X= 10-6) was calculated. From this, the critical cooling rate for glass formation is estimated as about 7 x 10 -2 °K sec-~ and the thickness of sample which can be formed as a glass as about 7 cm. Using measured values of the growth rate ts) increases the (dT/dt)¢ estimate by a factor of about 5 and decreases the y~ estimate by a factor of about 2. 3.3. SALOL In this case we shall use the viscosity data of Jantsch 19) and Laughlin and Uhlmann20, el) together with: TE=316.6 °K; a o = 6 /~; N ° = 1022 molecules c m - 3; AHfM= 2340 cal (g-atom)- 1 ; DT u = 5 X 10- 3 c m 2 sec- 1 ; and f given by eq. (7). The resulting T - T - T curves for volume fractions crystallized of l0 -6 and 10 -8 have been shown in fig. 1. The critical cooling rate for X = 10-6 is estimated as about 50 °K sec-1, and the thickness obtainable in the glassy state as about 0.07 cm. Using measured values of the growth rate e2) decreases (dT/dt)~ to about l0 °K sec -1 and increases y~ to about 0.15 cm. The importance of the melting point relative to the viscosity-temperature relation for a given type of material may graphically be illustrated by the case of salol. Fig. 3 shows T - T - T curves (X= 10- 6) calculated for materials having the same viscosity-temperature relation, entropy of fusion, and other properties as salol but with assumed melting points of 356.6, 316.6 and 276.6 °K, respectively. The melting point of salol is 316.6 °K, and the other assumed melting points represent temperatures 40°K above and below the actual melting point. The viscosities at the three assumed melting points are, respectively, about: 0.03, 0.08 and 1.5 poise. l

?

4.0

,~

6C

o: ul m

8C

g

~

I

I

I

I

]

I

I

t

1

]

I

t

tO -t

~o

to'

Lo~

,o ~

,o'

,o~

,o ~

~o'

~o~

Lo~

IZC

~o-'

io-3

tO-2

I

i

I

TIME

i

i

t

i

i

i

t

I

,d o

,o"

(SEC)

Fig. 3. T i m e - t e m p e r a t u r e - t r a n s f o r m a t i o n curves for salol-like materials h a v i n g v a r i o u s melting points. V o l u m e fractions crystallized o f 10- 6, (A) T~ := 356.6°K; (B) T~. = 316.6°K a n d (C) T~ = 276.6°K.

344

u.R. UHLMANN

It is apparent from fig. 3 that the glass-forming characteristics of materials vary strongly with the location of the melting point relative to the viscositytemperature relation. This variation, which has been suggested in different forms previously can now be expressed in a more quantitative manner. Specifically, the estimated cooling rates required to avoid detectable crystallization are, respectively, about: 105, 50, and 5 x 10-6 o K s e c - 1 ; and the corresponding thicknesses obtainable as glasses are about: 2 x 10- a, 7 x 10- 2 and 2 x 10 2 c m . 3.4. WATER In evaluating the glass-forming characteristics of water, we shall combine high-temperature viscosity data zS) with the glass transition temperature (140 °K) observed for vapor-deposited samples z4). Taking the viscosity at the glass transition as 1013 poise, and drawing a smooth curve between this point and the high-temperature data, the viscosity-temperature relation of 13

t

I

1

I

I

I

I

3.0

3.6

4.2

4.8

5,4

6.0

6.6

12--

II-I0-9 8 7 6

F"

4

2 I 0 -I -2 -3

I//T

7.2

x I0 3

Fig. 4. Hypothetical viscosity-temperature relation for water, based on high temperature viscosity data of ref. 12 and glass transition point of ref. 24.

345

KINETIC TREATMENT OF GLASS FORMATION

fig. 4 was constructed. Using this relation, together with T E = 273 °K, ao = 3A, N ° = 3 x 1022 molecules cm -3, AHfM= 1420 cal (g-atom) -1, DTH= 1.3 × 10 -3 cm 2 sec-1, and.f= 1, the appropriate T - T - T curve was obtained (X= 10-6). From this, the estimated cooling rate needed to avoid detectable crystallization is about 107 °K sec-1, and the corresponding thickness obtainable as a glass is about 1 ~tm. 3.5. METALS The metallic elements present a greater problem for estimating the critical conditions for glass formation. This may be seen by considering the case of silver, whose high-temperature viscosity behavior is reasonably well characterized25). Unfortunately, nothing is known about the glass transition temperature of this or any other pure metal. For such materials, amorphous 14

I

I

I

I

I

I

12

I0

go

6--

4.--

2--

0--

-2 5

I

I

I

I

I

I

I0

15

20

25

30

35

I/T

x 104

Fig. 5. Hypothetical viscosity-temperature relation for molten silver, based on high temperature viscosity data of ref. 25 and glass transition data on metal alloy glass of ref. 27.

346

D.R.UHLMANN

solids could not be obtained even by condensing from the vapor onto a substrate at liquid helium temperatures. The semi-metals Bi and Ga could be deposited as amorphous solids under these conditions, but were observed to crystallize at temperatures less than 15 °K 26). It should be noted, however, that unlike molecularly complex materials, crystallization of metals can take place without molecular reorientation, and the temperature dependence of the interface process can be quite different from that of transport in bulk liquid. In the present section we shall neglect the possibility of such crystallization without molecular reorientation, and will assume that the growth rate can be related to the viscosity by a relation of the form ofeq. (6). This is in keeping with our concern for the minimum cooling rate required for glass formation. Also in accord with this concern, we shall take the glass transition temperature of silver as that of a metallic Au-Si-Ge alloy z7), while recognizing that the actual Tg of the elemental metal may be appreciably lower. Taking, then, r/= 1013 poise at T=285 °K together with the high-temperature viscosity data cited previously, the viscosity-temperature relation shown in fig. 5 was obtained. Using this relation, together with TE=1234°K, a o = 2 . 7 A , NO= 1.I x 1022 atoms cm -3, AHem=2730 cal (g-atom) -1, DTa= 10 -1 cm 2 sec-2, and f = 1, the appropriate T - T - T curve was constructed for X = 10-6. From this, the estimated cooling rate needed to avoid detectable crystallization is about 1020 °K sec -2, and the corresponding thickness obtainable as a glass is about 1000 A. 4. Discussion and conclusions

In the preceding sections, we have adopted the view that the critical conditions (cooling rate and thickness) for forming a glass of a given material can be estimated from even rather limited information about the material. The critical conditions are estimated from time-temperature-transformation (T-T-T) curves, taking a volume fraction crystallized of about 10- 6 as the minimum detectable crystallinity. Our concern has been directed to the minimum conditions for glass formation, and to the insight which they provide into the glass-forming abilities of various materials. We have then considered only homogeneous nucleation and a volume fraction of crystals which must be avoided for glass formation. The present approach may readily be generalized to include heterogeneous nucleation in the volume of the liquid or on the external surfaces, or diffusion-controlled growth rates, or practical heat-flow situations; and these extensions are presently being carried out~8).

KINETIC TREATMENT OF GLASS FORMATION

347

The greatest uncertainty in the present calculations lies in the estimates of the nucleation frequencies. From the limited available data on nucleation in condensed phases, these estimates should not be taken as accurate to better than some five to eight orders of magnitude (corresponding to one or two orders of magnitude in the calculated critical cooling rates). The view has here been adopted that nearly any liquid is a potential glassformer, but that large variations exist in the cooling rates which must be employed to obtain specimens as amorphous rather than crystalline solids. In the case of the network glass-formers, SiO2 and GeO2, the cooling rates required for glass formation are well in the range of standard processing techniques; and little difficulty is encountered in forming them as glasses, even when allowance is made for the presence of nucleating heterogeneities. As indicated by the estimated cooling rates, SiO2 is a much better glass-former than GeO z; and both can more readily be formed as glasses than most simple organic liquids. In the case of H20, an additional uncertainty exists in the estimated cooling rate because of the wide interpolation used with the viscosity data. Despite the related uncertainty, however, the calculated minimum cooling rate is sufficiently close to the maximum attainable with any material that the formation of glassy water by cooling the melt must be regarded as a doubtful proposition. In the case of metallic Ag, a wide extrapolation in viscosity was again made, but the estimated cooling rate should be well on the low side because of the high glass transition temperature assumed. Hence, since the calculated minimum cooling rate exceeds that obtainable with any known technique, the formation of glassy pure metals from the liquid phase must be regarded as highly unlikely. Extrapolating from these results, we can conclude that the most favorable conditions for glass formation involve a large viscosity at the melting point of the crystalline phase and/or a rapidly rising viscosity with falling temperature below the melting point. The important oxide glass-formers are generally characterized by the former, while organic glass-formers are usually marked by a large Idq/dT]. The importance to glass formation of a low melting point relative to a given viscosity-temperature relation has been illustrated by the estimates for salol-like materials (a low TE/Tgimplies for a given class of materials a relatively large r/ at TE and a relatively large [dq/dTI). The importance of near-eutectic compositions in favoring glass formation in alloy systems is related to these same considerations - and also to the redistribution of material required for crystallization to proceed in cases where a single crystalline phase is not stable relative to the amorphous phase. The dependence of the critical cooling rate on the location of the melting point for salol-like materials should be fairly representative of many organic liquids. For most oxide systems, however, a much less pronounced sensitivity

348

D.R. UHLMANN

is expected. This is related to the smaller t e m p e r a t u r e d e p e n d e n c e o f the viscosity o f oxide liquids c o m p a r e d with t h a t o f organic liquids.

Acknowledgments F i n a n c i a l s u p p o r t for the present w o r k has been p r o v i d e d by the N a t i o n a l A e r o n a u t i c s a n d Space A d m i n i s t r a t i o n . This s u p p o r t is gratefully a c k n o w ledged.

References 1) D. Turnbull and M. H. Cohen, J. Chem. Phys. 29 (1958) 1049. 2) D. Turnbull and M. H. Cohen, J. Chem. Phys. 34 (1961) 120. 3) W. B. Hillig, in: Symposium on Nucleation and Crystallization in Glasses and Melts (Am. Ceramic Soc., Columbus, Ohio, 1962). 4) D. R. Uhlmann, in: Materials Science Research, Vol. 4 (Plenum, New York, 1969). 5) W. B. Hillig, in: Reactivity of Solids (Wiley, New York, 1969). 6) M. H. Cohen and D. Turnbull, J. Chem. Phys. 31 (1959) 1164. 7) G. Adam and J. H. Gibbs, J. Chem. Phys. 43 (1965) 139. 8) D. Turnbull, Contemp. Phys. 10 (1969) 473. 9) W. A. Johnson and R. F. Mehl, Trans. AIME 135 (1939) 416. 10) M. Avrami, J. Chem. Phys. 7 (1939) 1103. 11) M. Avrami, J. Chem. Phys. 8 (1940) 212. 12) M. Avrami, J. Chem. Phys. 9 (1941) 177. 13) J. W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, New York, 1965). 14) K. A. Jackson, in: Growth and Perfection of Crystals (Wiley, New York, 1958). 15) K. A. Jackson, in: Progress in Solid State Chemistry, Vol. 4 (Pergamon, New York, 1969). 16) E. H. Fontana and W. A. Hummer, Phys. Chem. Glasses 7 (1966) 139. 17) F. E. Wagstaff, J. Am. Ceram. Soc. 52 (1969) 650. 18) P. J. Vergano and D. R. Uhlmann, in: Proc. VIlth Intern. Syrup. on the Reactivity of Solids (Wiley, New York, 1969). 19) O. Jantsch, Z. Krist. 108 (1956) 185. 20) W. T. Laughlin, Viscous Flow and Volume Relaxation in Simple Glass-Forming Liquids, Sc.D. Thesis, Mass. Inst. of Tech., Cambridge, Mass., 1969. 21) W. T. Laughlin and D. R. Uhlmann, The Viscous Flow of Simple Glass-Forming Liquids, to be published. See Bull. Am. Ceram. Soc. 47 (1968) 402. 22) K. A. Jackson, D. R. Uhlmann and J. D. Hunt, J. Crystal Growth 1 (1967) 1. 23) R. C. Weast, ed., Handbook of Chemistry and Physics (Chemical Rubber Co., Cleveland, Ohio, 1964). 24) J. A. McMillan and S. C. Los, Nature 206 (1965) 806. 25) C. J. Smithels, ed., Metals Reference Book (Wiley, New York, 1955). 26) R. Hilsch, in: Non-Crystalline Solids (Wiley, New York, 1960). 27) H. S. Chen and D. Turnbull, J. Chem. Phys. 48 (1968) 2560. 28) G. Scherer and D. R. Uhlmann, to be published.