A kinetic treatment of glass formation VII: Transient nucleation in non-isothermal crystallization during cooling

Journal of Non-Crystalline Solids 50 (1982) 189-202 North-Holland Publishing Company

A KINETIC TREATMENT OF GLASS FORMATION TRANSIENT NUCLEATION IN NON-ISOTHERMAL CRYSTALLIZATION DURING COOLING H. YINNON

189

VII:

and D.R. UHLMANN

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 15 September 1981

The analysis of crystallization statistics has been modified to allow for time-dependent (transient) nucleation. To establish its accuracy, the numerical analysis has been applied to isothermal crystallization kinetics and shown to yield crystallization versus time curves which compare very closely with curves calculated analytically with or without the inclusion of transient nucleation. The numerical analysis including transients has been used to calculate the critical cooling rates for glass formation in anorthite and o-terphenyl considering (1) only homogeneous nucleation and (2) homogeneous nucleation+heterogeneous nucleation for 107 heterogeneities cm -3 with contact angles between 40 ° and 100°. It has been shown that inclusion of time-dependent nucleation in the calculations does not change the critical cooling rates for glass formation calculated assuming steady-state homogeneous nucleation in both materials. The critical cooling rate in anorthite calculated including steady-state heterogeneous nucleation was found to be decreased only slightly by the inclusion of time-dependent nucleation; while the critical cooling rates calculated for o-terphenyl were not changed at all by the inclusion of time-dependent nucleation. The lack of an effect of time-dependent nucleation on the critical cooling rates calculated assuming only homogeneous nucleation is explained by the relatively small transient times on the high temperature side of the nucleation peak (a temperature range which has an overwhelming effect on the overall crystallization process because of the relatively high crystal growth rates in this range). Although the critical cooling rates associated with heterogeneous nucleation are large, the nucleation here takes place at relatively small undercoolings where the transient times are relatively small. Thus transient nucleation causes only a temporary delay in the overall crystallization, and its effect on the critical cooling rate is small.

1. Introduction F r o m t h e p e r s p e c t i v e o f k i n e t i c t r e a t m e n t s of glass f o r m a t i o n (e.g. refs. 1 a n d 2), t h e c r i t i c a l q u e s t i o n in f o r m i n g a glass is c o o l i n g at a s u f f i c i e n t l y h i g h r a t e t h a t d e t e c t a b l e c r y s t a l l i z a t i o n is a v o i d e d . I n a d d r e s s i n g this issue, use is g e n e r a l l y m a d e o f the f o r m a l t h e o r y o f t r a n s f o r m a t i o n k i n e t i c s o r i g i n a l l y d e v e l o p e d b y J o h n s o n a n d M e h l [3] a n d A v r a m i [4]. T h e s e w o r k e r s o b t a i n e d a n a l y t i c a l e x p r e s s i o n s f o r the v o l u m e f r a c t i o n c r y s t a l l i z e d as a f u n c t i o n o f t i m e 0022/3093/82/0000-0000/$02.75

© 1982 N o r t h - H o l l a n d

190

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

under isothermal conditions where either steady state or time dependent nucleation prevails. For crystallization under non-isothermal conditions, appropriate when a liquid is cooled to form a glass, the approach originally suggested by Grange and Kiefer [5] or a variant upon this approach, is widely used [6]. Such crystallization has recently been shown by Onorato et al. [7] and by Yinnon and Uhlmann [8] to be amenable to numerical solution by the technique of crystallization statistics, which simulates the detailed crystallization process of a supercooled liquid as its temperature is changed. Utilizing available data on crystal growth rates, liquid viscosity and nucleation frequency, the technique has been used to simulate crystallization upon reheating a glass in DTA experiments [7]. The technique has also been used to explore the effects of heterogeneous nucleation on the critical cooling rate required to form a glass, and to indicate the type of behavior expected in a nucleation experiment when several types of nucleating heterogeneities are present [8]. In these and other analyses carried out to date, the treatment of crystallization statistics has assumed that the nucleation frequency depends only on temperature. In the present paper, a modification of the treatment is described which takes into account the time dependence of the nucleation frequency. The approach described here is specifically aimed at describing the effects of transient nucleation on the overall crystallization during the cooling of a supercooled liquid, and thus the effects on the critical cooling rates required to form glasses. The approach can also be used in assessing the effects of time dependent nucleation in other experimental conditions, e.g. crystallization upon reheating a glass in DTA experiments.

2. Time dependent nucleation The process of crystal nucleation and the conditions under which time-dependent nucleation prevails have been described by Gutzow and Toschev [9], Christian [6] and more recently by Russell [10] and Yinnon and Uhlmann [8]. The dependence of the nucleation rate on time results from the requirement of achieving the steady-state distribution of embryos of the crystalline phase within the liquid, a process which requires some finite time. Assuming that a liquid is quenched infinitely rapidly from above its melting point to some temperature T where it is held, the distribution of crystal embryos changes with time to reflect the changed conditions. The distribution reached at equilibrium is: Nfi = N° e x p ( - A G R / k T ) ,

(1)

where Nfi is the number of embryos of radius R per unit volume; N° is the number of molecules per unit volume, and AGR is the free energy of formation of an embryo of size R. As noted originally by Becker and Dbring [ 11], the equilibrium distribution

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation Vll

191

of eq. (1) does not consider that supercritical embryos may also shrink again (although they are more likely to grow). The steady-state distribution, which takes account of the forward and back fluxes for each size of embryo, is very close to the equilibrium distribution for small R. It decreases progressively below the equilibrium distribution with increasing R; is half that predicted by eq. (1) for the critical size; and is continuous through the critical size, approaching zero for R notably larger than the critical size. Use of the steady-state distribution results in a calculated nucleation rate with the same exponential factor as that obtained using the equilibrium distribution, but with a modified (reduced) pre-exponential factor. The nucleation frequency is related to the rate at which embryos of critical size are formed. Hence it is small at the beginning of the isothermal hold and increases until the steady state distribution is established. The time dependence of such isothermal nucleation is evaluated from the kinetics of embryo formation [6,17], and involves a characteristic time, ~', termed the transient or incubation time. In terms of ~', the time-dependent isothermal nucleation frequency, Iv(t ), is usually expressed by the approximate relation [13]:

I~(t) = Iv(sS ) e - ' / ' ,

(2)

where Iv(ss ) is the steady-state nucleation frequency. As noted by Christian [6], this expression may provide a reasonable approximation to the nucleation rate in the early part of the transient. The transient time is given to order-of-magnitude accuracy by:

(3)

•

where n* is the number of molecules in the critical size nucleus; Ns is the number of surface molecules on the nucleus of critical size, and p is the frequency factor for transport at the nucleus-liquid interface. For molecularly complex materials, such as most organics and inorganic oxides, the temperature dependence of 1, can usually be taken as =

(4)

where ~ is the viscosity; and b is given by:

kT bOROANICS ~

3~ra° ,

(5A)

lOkT bSILICATEs ~

3era ° .

(5B)

Here a 0 is a molecular diameter. Eq. (5A) represents the familiar Stokes-Einstein relation; and eq. (5B) is an empirical relation based on crystal growth data [14]. More detailed treatments of time-dependent nucleation have been provided by Kantrowitz [15], Probstein [16], as well as by Turnbull [12], Russell [10] and

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

192

Kashchiev [17]. No one provides an exact description of the process, valid over the full range of times. A simpler expression than eq. (2) is sometimes used to describe time-dependent nucleation: Iv(t ) = 0 ,

for t<~-

Iv(t)=Iv(SS)(1-b~ ),

fort>T

(6)

where b = ,r 2/6. The formal theory of transformation kinetics was modified by Gutzow and Kashchiev [18] to include the effects of transients on isothermal crystallization. Using eq. (6) to represent the nucleation rate, the volume fraction crystallized was expressed as:

Vc/V= 0 for t ~ b % where Vo/V is the volume fraction crystallized at time t and

(7)

where the growth is assumed to be isotropic and independent of time. In the non-isothermal case of a continuous cooling experiment, the steady state distribution of embryos changes with time, as does the transient time ~(which is a function of viscosity and size of the critical embryo). If one assumes that the cooling is carried out stepwise, such that the liquid is held isothermally for some time interval At at successively decreasing temperatures, the steady state distributions of embryos can be assigned to each time interval as shown in fig. 1.

-2

I

I

I

I

I

I

-4

-6 > -8 Z~ _

(R

_~ -I0 -12

-14 -16 I0

t478 II

12

13

14

R(1)

15

16

17

Fig. 1. Steady state distributions of embryos calculated for anorthite at temperatures near the m a x i m u m of the homogeneous nucleation rate. The temperatures indicated are the temperatures of successive time intervals used in the crystallization statistics calculations. For further details see text.

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

The size of the critical nucleus at each temperature is designated R* in fig. 1; and the number of such critical embryos N7 is reflected in the nucleation rate. Cooling from temperature T I to temperature T2 decreases the value of R* and increases N*. Immediately after cooling from T 1 to T2, all nuclei with radii between R~ and R~ become supercritical and contribute to an instantaneous increase in the nucleation frequency, a phenomenon termed athermal nucleation [19]. The concentration of critical embryos of radius R~ then rises with time from its steady-state value at T I [NI(R~) ] to its steady-state value at T2( N~ ). These considerations will be used in the following section to evaluate the effects of nucleation transient times on overall crystallization behavior during continuous cooling.

3. Crystallization during continuous cooling In evaluating the effects of transient nucleation on the crystallization of a liquid during continuous cooling, the following assumptions are made. (1) The temperature difference between two successive time intervals in the cooling calculations is sufficiently small that the change of the stable embryo size between intervals is also small. Hence the number of embryos that instantaneously become supercritical upon cooling from temperature T~ to temperature T z is negligible - i.e., athermal nucleation can be ignored. The validity of this assumption can be examined by considering fig. 1, which shows embryo distributions in anorthite at temperatures near the maximum in rate of homogeneous nucleation for the time intervals used in the calculations. The temperature difference between two time intervals is 14 K; and as can be seen from the figure, the change in R* is negligible for such changes in temperature. In the temperature range of maximum heterogeneous nucleation (for contact angle0 = 40 °, approximately 1700 K), the change in R* between time intervals is approximately 5 ,~. (2) Upon cooling from T 1 to T2, the steady-state number of critical embryos at T2, N~, is much greater than the steady-state number of embryos of the same radius, R~, in the distribution at TI. This assumption enables us to use eqs. (2) or (6) to express approximately the time dependence of the nucleation frequency (see below), since it can be assumed that the stable embryos of size R~ build up effectively from zero concentration. (3) The initial rate of nucleation at temperature T2 should correspond to the steady state concentration at temperature T~ of embryos having a radius of R~. This initial nucleation frequency at T z is different from the steady-state nucleation frequency at T 1 which corresponds to R~'. Because of the small difference between R~' and R~, however, the difference between these rates is small. It will therefore be assumed that the initial nucleation frequency at T2 is the steady-state nucleation frequency at T I subject only to the reservation in Assumption (4) below.

193

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

194

(4) So far, it has been assumed that the time interval at T 1 is longer than T (i.e., At >>~-), so that the nucleation frequency at the end of the time interval at T 1 is the steady-state nucleation frequency at that temperature. If this is not the case, the initial nucleation frequency at T2 is taken to be the final nucleation frequency achieved during the time interval at T r

3.1. The numerical method The details of the crystallization statistics treatment are given elsewhere [7,8]. The time that the liquid is being cooled from the liquidus temperature to the glass transition temperature is divided into 50 intervals within which the crystallization is treated isothermally. The numbers and sizes of the nucleating crystallites are calculated at each time interval and the total crystallized volume is calculated by summing over all these crystallites, allowing for impingement according to the treatment by Avrami [4]. To determine how well the numerical solution compares with the analytical solution available for the isothermal case, the crystallization statistics program was used assuming constant temperature. The Avrami equation describing such a case for isotropic, time independent growth with a constant nucleation frequency is given as:

Vc/V= 1 -- exp(--Kt4),

(8)

where K is a constant which includes the nucleation frequency and the crystal growth rate. A plot of log[-ln(1 - VJV)] versus log(t) should be a straight line with a slope of 4. Fig. 2 shows such a plot for o-terphenyl where the values of t and Vc/V were calculated by the crystallization statistics analysis. For Vc/V> 10 -2 a straight line with a slope of 3.99 is obtained indicating an excellent agreement between the numerical method and eq. (8). A smaller value I

I

I

o-Terphenyl

/

0

I

~-I

-2

2.0

i

I

2.2

I

I

24

i

I

2.6

log ( t ) Fig. 2. Volume fraction crystallized versus time relation for o-terphenyl testing form of eq. (8). Volume fraction crystallized calculated using the numerical analysis of crystallization statistics. The time interval, At, used in the calculations is 5 min; the slope is 3.99.

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

195

is obtained for the slope for smaller Ixc/Ix values and when excessively long time intervals are used. In order to allow for transient nucleation, the value of r is calculated at each time interval. If for any time interval 0-< At/100, the interval At is further divided into 20 intervals; and at each such interval the nucleation frequency is calculated as: I v ( t ) = Iv(1 ) + [Iv(2) - Iv(l)] e x p ( - 0 " / t ) ,

(9)

where Iv(1 ) is the nucleation frequency at the end of the time interval corresponding to T, and I v (2) is the steady state nucleation frequency at T2. Eq. (9) assumes a Zeldovitch-type time dependence of I v, and is based on the assumptions detailed above. An average value of the nucleation frequency is calculated from the 20 values thus obtained, and this serves as the nucleation frequency for the time interval At. For the next time interval, the nucleation frequency, Iv(1 ) is taken as the m a x i m u m value of the nucleation frequency obtained previously and not as the average. Fig. 3 shows a graphical representation of the time (temperature) dependence of the nucleation frequency for anorthite containing 107 heterogeneities cm -3 characterized by a contact angle of 60 °, cooled at 2.3 X 106 K m i n - ' when transient nucleation is taken into account. As the temperature decreases, the value of the transient time, ~', increases until it becomes much greater than At, after which the nucleation frequency ceases to change. 1:5 Anorthite

II

Nv = 107 c m - 3 O = 60* Rc = 2 " 3 x 1 0 6 K m i n ' l

9 7

....../ f

Iv(ss)'4 ,-,>5 o

,,,....../

,a..~........ ,........

e-'"''' ,,'""'

~'--Iv(t),averaged over

a

time intervol&t

..,.......

J ,,.....'" ~ " I v ( t )

1.5

A t : 6.1 .lO'5min i r

O-";..a -:5

-5

ffr 1

1660

1.0
r/At I

1632

1

I

1604 1576 Temperature (K)

I

1548

-0.5

I

1520

~"

0

Fig. 3. Temperature (time) dependence of the steady state heterogeneous nucleation frequency, lv(ss), time dependent heterogeneous nucleation frequency, Iv(t ), and the average over a time interval At for the time-dependent nucleation frequency for anorthite containing 107 heterogeneit i e s c m - 3 characterized by 0 =60 ° and cooled at the critical cooling rate of 2.3X 10 6 K rain ~. The ratio of the transient time, ~-, to the time interval, At, is also shown.

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

196

l.OO-

Anorthite T = 1200K At = 10 min 0.75 -

T ii 21 min

> ‘0.!50>”

0.25-

0 Time --+ Fig. 4. Isothermal progression of V,/ V with time for anorthite at 1200 K when only steady state nucleation is considered [I,(U)] and when transient nucleation is considered [I,(t)]. The curve corresponding to I,(t) is displaced by -40 min with regard to the curve corresponding to I,( SS).

The development of the volume fraction crystallized, V,/V, can be modelled using the treatment of crystallization statistics at a nearly fixed temperature, and thus the numerical method can be compared with an analytical solution for isothermal crystallization with transient nucleation, e.g., eq. (7). The expression of eq. (7) is based on a time dependence of the nucleation frequency having the form of eq. (6). While eq. (2) is assumed in the present work, the time dependence of the volume fraction crystallized is not expected to depend critically on the actual expression used for I,(t). A comparison of eq. (7) with eq. (8) shows that the curve of V,/V versus time is shifted by a time constant of b7 when transient nucleation is included. This behavior is observed in fig. 4 where the calculated V,/V is plotted versus time for anorthite at 1200 K both when transient nucleation is included and when steady-state nucleation only is assumed in the treatment of crystallization statistics. The curve calculated for transient nucleation is displaced by approximately 40 min relative to the curve calculated for steady-state nucleation. The value of about 40 min compares well with the expected value based on eq. (7) of 34 min, thus indicating that the model used in the numerical calculations successfully predicts the development of the crystallized volume in an isothermal situation.

4. Transient nucleation and glass formation The description of crystallization during continuous cooling is of interest in estimating the critical cooling rate necessary to form a glass, i.e., the slowest

1t. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

197

rate at which a liquid can be cooled between its melting point and glass transition temperature without developing a detectable degree of crystallinity. The most familiar technique for evaluating the critical cooling rate involves calculating curves which represent the locus of the temperatures and times at which the just-detectable degree of crystallinity is developed at various cooling rates. Such curves are known as continuous cooling or CT curves, and can be obtained from the corresponding time-temperature-transformation (TTT) curves using the approach of Grange and Kiefer [5]. For the application of this approach to the problem of glass formation, see ref. 20. Yinnon and Uhlmann [8] applied the analysis of crystallization statistics to two well-characterized liquids, anorthite (CaO. A1203- 2 SiO2) and o-terphenyl, as representative of inorganic oxide and simple organic glass-forming liquids. These authors investigated the effects on critical cooling rates of various concentrations of nucleating heterogeneities characterized by different contact angles. It was found that with decreasing contact angle, the curves describing crystallization during continuous cooling shift to shorter times and higher temperatures, and the critical cooling rates increase. The same two materials, anorthite and o-terpehnyl, are considered in the present evaluation of time-dependent nucleation and its effect on the cooling 3

6[ O-Terphenyl C4F

?_,

/

~

[

21-

Anorthite I- ~neous

I..,

_2~'1

I

I

I

/ -

RC--220 K rnin 4

-3

I

80

Iv(SS

nucleation

/

I

_

_

0.08 0.06 .~_

0.04 E E

4) 2

265

255

245

Temperature(K)

235

=

1380

1310

•

"

1240

Temperature(K)

0.02 0 170

Fig. 5. Steady state homogeneous nucleation frequency, Iv(ss), actual nucleation frequency (including transient nucleation), Iv(t), crystal growth rate, u, and log~o(~/At ) in o-terphenyl cooled at the critical cooling rate of 0.6 K r a i n - t. Fig. 6, Steady state homogeneous nucleation frequency, lv(ss), actual nucleation frequency including transient nucleation), Iv(t ), crystal growth rate, u, and log(T/At) in anorthite cooled at the critical cooling rate of 220 K n'fin- 1.

198

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

rates required to form glasses. The calculations indicate that although the ratio of r/At may well exceed unity in some time intervals during cooling of the two representative liquids, the overall rates of crystal nucleation and growth - and thus the critical cooling rate - are not greatly affected by the introduction of time-dependent nucleation rates in the calculations. Since the origin of this conclusion is somewhat different for homogeneous nucleation than for heterogeneous nucleation, continuous cooling with the two types of nucleation will be discussed separately. Fig. 5 shows the steady-state nucleation frequency, the actual nucleation frequency calculated as outlined in the previous section, the crystal growth rate, and the ratio ,r/At for o-terphenyl containing no heterogeneities cooled at 0.6 K min-~, the critical cooling rate for glass formation. The same critical cooling rate is obtained when either steady-state nucleation or time-dependent nucleation behavior is assumed. At the maximum of the steady-state nucleation peak ( T = 251 K), the ratio "fiAt is 45; and the actual nucleation frequency is approximately 30% of the steady-state nucleation frequency. The origin of the small effect of transients on the critical cooling rates for glass formation lies in the pronounced temperature dependence of the transient time. For example, at a temperature only 10 K above that of the maximum nucleation rate, "fiAt is only 0.25; and at higher temperatures (where the growth rate is high), the ratio is even smaller. Hence over nearly all of the temperature range where the overall kinetics of crystallization are most rapid, there is only a small difference between the actual nucleation frequency (including transient effects) and the steady-state nucleation frequency. The contribution to the overall extent of crystallization of the nuclei formed at the temperature of the peak nucleation rate and lower temperatures is very small since the crystal growth rate at those temperatures is rapidly decreasing with continued cooling (see fig. 5). The same remarks are also applicable to anorthite; although for this material, the ratio z/At at the peak in nucleation rate is only about 2.2. Hence, as shown in fig. 6, the actual nucleation frequency, including the effects of transients at the critical cooling rate of 220 K min-~, deviates by only a very small amount from the steady-state frequency. Fig. 7 shows the logarithms of the calculated steady-state and actual (including transients) nucleation frequencies, the logarithm of the crystal growth rate as well as the ratio ,r/At in anorthite containing 107 heterogeneities c m - 3 with a contact angle of 60 ° cooled at the critical cooling rate of 2.3 × 10 6. The critical cooling rate without transient nucleation is 2.7 X 10 6 K m i n - ~. At such a cooling rate, the overall crystallization process is completely dominated by the heterogeneous nucleation which occurs in the range of small undercoolings. The time interval At is much smaller than ~ at this cooling rate in the temperature range of homogeneous nucleation. Since, however, heterogeneous nucleation takes place at higher temperatures (see fig. 4 in ref. 8), the ratio of z/At is modest even at very fast cooling rates over the temperature range important in crystallization. Thus, as shown in fig. 7, the actual nucleation

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII 0.4~-

3

O-Terphenyl

199

/

Anorthil6 --

2

0 ~ " - r ~

0 12

I

1

I

0.1 0

-~ E -0.1 E

v> 2

-0.2 3 -05 ~

-2 At=e.,,lO'Sm,.

-4

-6

I

1660

I

I

8

~'8 E 9 6 E c~ ._c 4

J

I

I0-

I

Iv( s s ) - . . . ~ -

10

I

\ I

1590 1520 Temperoture ( K )

"0.4 ~' I

1450

0.5

ss)

g4

1.2 14T"

_c

E

c_ 2

g

Z>o

1.6 c_ 3

--1

-4

1.8 ~

-6 -8

At =2F~10-4mm I

510

304

I

I

\ I

298 292 286 Temperoture (K)

~.

2.0 280

Fig. 7. Logarithms of the steady state heterogeneous nucleation frequency, ljss), the actual nucleation frequency (including transient nucleation), Iv(t ), and the crystal growth rate, u, as well as the ratio r/At in anorthite containing 107 heterogeneities cm -3 with 0 =60 ° cooled at the critical cooling rate of 2.3X 106 K min-t Fig. 8. Logarithms of the steady state heterogeneous nucleation frequency, lv(ss), the actual nucleation frequency (including transient nucleation, Iv(t), and the crystal growth rate, u, as well as the ratio "r/At in o-terphenyl containing 107 heterogeneities cm -3 with 0 =60 ° cooled at the critical cooling rate of 104 K min i. frequency is close to the steady-state frequency. Fig. 8 shows the calculated logarithms of the steady-state nucleation frequency, the actual nucleation frequency (including transients), the crystal growth rate and the ratio "r/At for a sample of o-terphenyl containing 107 nucleating heterogeneities cm -3 with a contact angle of 60 ° which is cooled at the critical cooling rate of 104 K r a i n - ~. The ratio r / A t at the m a x i m u m of the steady state nucleation frequency is only 0.25. Hence the actual nucleation frequency is very close to the steady-state value, and the critical rate calculated for o-terphenyl is not affected b y transient nucleation. The critical cooling rates calculated for o-terphenyl and anorthite are shown in table 1. The inclusion of transient nucleation affects only the critical cooling rates in anorthite containing 107 nucleating heterogeneities with a wetting angle, 8, smaller than 90°; and even in these cases, the effect of transients is only modest. Continuous-cooling curves can also be constructed for a given liquid containing a given concentration of heterogeneities with wetting angle, O, for different values of Vc/V. Such curves are shown in fig. 9 for anorthite containing 107 nucleating heterogeneities cm -3 with 0 = 60 ° for V c / V values of 10 -2,

200

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation VII

1700

I

I

I

Anorthite ..... 1 6 0 0

Y

/'/,'

//"

1500

o_ E

i

lO. 2

O= 6 0 * Nv = I 0 "t cm "3

1400

Iv (ss) - Iv(t) 1500 -5

I -4

I -5

....

I -2

-I

Loglo(t) Fig. 9. Continuous cooling curves for anorthite containing 107 heterogeneities cm -3 with 0 - - 6 0 ° calculated for V c / V values of 10 -2, 10 -6 and 10 - I ° . Curves drawn in full lines are calculated assuming steady state nucleation only, while curves drawn in dashed lines are calculated allowing for transient nucleation.

10 .6 and 10 -1°. A critical cooling rate of 1.8 × 105 K min -1 is calculated for Vc/V = 10-2, whether transients are included or neglected. Hence crystallization during continuous cooling is not affected at all by transient nucleation for this volume fraction crystallized. For the widely-used just detectable degree of crystallinity (Vc/V= 10-6), Rc is calculated as 2.25 × 10 6 K rain -1 when steady-state nucleation is assumed. At such a cooling rate, the actual nucleation frequency (including transients) deviates somewhat from the steady-state value; and the continuous cooling curve calculated using the actual nucleation frequency [Iv(t)] deviates somewhat from the one calculated using steady-state nucleation. The difference between the two curves is quite small, however, as shown in fig. 9; and hence transients do not have a sizable effect on the critical cooling rates required to form glasses of oxides and organic materials. For a much smaller volume fraction crystallized of 10-10, a critical cooling rate of 1.1 X 107K min - l is indicated if steady-state nucleation is assumed. At such a large cooling rate, the actual nucleation frequency deviates considerably from the steady-state value; and thus a continuous cooling curve which takes into account time-dependent nucleation deviates considerably from that calculated assuming steady-state nucleation.

5. Summary and conclusions The analysis of crystallization statistics has been modified to allow for time-dependent nucleation. It has been shown that the numerical analysis of crystallization statistics compares favorably with isothermal analytical solutions obtained with and without the inclusion of transient nucleation.

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation Vll

201

Table 1 Critical cooling rates, Re, calculated for anorthite and o-terphenyl containing 107 nucleating heterogeneities cm 3 with contact angles 0 0

40 ° 50 ° 60 ° 70 ° 80 ° 90 ° 100 ° 180 °

Anorthite

o-terphenyl

Steady-state nucleation R ~ ( K m i n 1)

Time-dependent nucleation R ~ ( K m i n l)

Steady-state and timedependent nucleation Rz(Kmin-l)

8.4X 6.7X 2.7X 3.2X 2.8X 2.3 X 3.0X 2.2X

7.7X 6.3X 2.3X 2.9X 2.7X 2.3 × 3.0X 2.2X

5.5 X 3.5X 1.0X 1.0X 7.0X 5.5 0.7 0.6

106 106 106 105 104 103 10 z 102

106 106 106 105 104 103 102 102

104 104 104 103 10 ~

The analysis of crystallization statistics has also been used to calculate the critical cooling rates for glass formation in anorthite and o-terphenyl containing 107 nucleating heterogeneities cm -3 with contact angles between 40 ° and 100 °. It has been shown that inclusion of time-dependent nucleation in the calculation did not change the critical cooling rates for glass formation calculated assuming steady-state homogeneous nucleation in both materials. The critical cooling rate in anorthite calculated for steady-state heterogeneous nucleation was found to decrease only slightly by the inclusion of time-dependent nucleation; while the critical cooling rates calculated for o-terphenyl were not changed at all by the inclusion of time-dependent nucleation. The lack of an effect of time-dependent nucleation on the critical cooling rates calculated assuming only homogeneous nucleation is explained by the relatively small T/At ratio at the high temperature side of the nucleation peak. This region of temperature has an overwhelming effect on the overall crystallization process because of the relatively high crystal growth rates in this temperature range. Heterogeneous nucleation always takes place at smaller undercoolings (higher temperatures) than homogeneous nucleation. In the region of higher temperature, the transient time, ~-, is small compared with the time interval At and thus transient nucleation causes only a temporary delay in the overall crystallization. Since the crystal growth rates at these high temperatures are high, the overall effect on the critical cooling rate is small even at cooling rates as large a s 10 7 K min -1. The small effects on calculated critical cooling rates, observed when transient nucleation is introduced in the analyses of crystallization statistics, indicate that the results obtained previously without considering transient nucleation for anorthite and o-terphenyl do not require significant modifi-

202

tl. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation 1Ill

cation. Other glass forming systems (such as metals) may, however, be affected more strongly by time-dependent nucleation, and such nucleation may be important in crystallization On reheating a glass. The modified analysis of crystallization statistics is capable of estimating such effects for any system for which viscosity and crystal growth data are available. Financial support for the present work was provided by the National Aeronautics and Space Administration. This support is gratefully acknowledged, as are stimulating discussions with Professor K.C. Russell of MIT.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

D.R. Uhlmann, J. Non-Crystalline Solids 7 (1972) 337. D.R. Uhlmann: J. Non-Crystalline Solids 25 (1977) 42. W.A. Johnson and K.F. Mehl, Trans. AIME 135 (1939) 416. M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941) 117. R. Grange and J.M. Kiefer, Trans. ASM 29 (1941) 85. J.W. Christian, The theory of transformations in metals and alloys, 2nd ed. (Pergamon, New York, 1975). P.I.K. Onorato, D.R. Uhlmann and R.W. Hopper, J. Non-Crystalline Solids 41 (1980) 189. H. Yinnon and D.R. Uhlmann, J. Non-Crystalline Solids 44 (1981) 37. I. Gutzow and S. Toschev, in: Advances in nucleation and crystallization in glasses, eds., L.L. Hench and S.W. Freiman (American Ceramic Society, Columbus, 1972). K.C. Russell, Adv. Coll. Interf. Sci. 13 (1980) 205. B. Becker and W. D/Sring, Ann. Phys. 24 (1935) 719. D. Turnbull, Trans. AIME 175 (1948) 774. J.B. Zeldovitch, Acta Physicochim. URSS 18 (1943) 17. G.W. Scherer and D.R. Uhlmann, J. Cryst. Growth 29 (1975) 12. A. Kantrowitz, J. Chem. Phys. 19 (1951) 1097. R.F. Probstein, J. Chem. Phys. 19 (1951) 619. D. Kashchiev, Surf. Sci. 14 (1969) 209. I. Gutzow and D. Kashchiev, in: Advances in nucleation and crystallization in glasses, eds., L.L. Hench and S.W. Freiman (American Ceramic Society, Columbus, 1972). J.C. Fisher, J.H. Hollomon and D. Turnbull, J. Appl. Phys. 19 (1948) 775. P.I.K. Onorato and D.R. Uhlmann, J. Non-Crystalline Solids 22 (1976) 367.