A kinetic treatment of glass formation V: Surface and bulk heterogeneous nucleation

Journal of Non-Crystalline Solids 44 (1981) 37-55 North-Holland Publishing Company

37

A KINETIC TREATMENT OF GLASS FORMATION V: S U R F A C E A N D B U L K H E T E R O G E N E O U S N U C L E A T I O N H. Y I N N O N and D.R. U H L M A N N

Department of Materials Science and Engineering, Massachusetts Institute o f Technology, Cambridge, Mass. 02139, USA Received 7 October 1980

An analytical method is described for calculating the detailed distribution of crystallite sizes in a supercooled liquid, and the changes in this distribution as a function of temperature (time) while the liquid is cooled from above the melting point. This method, termed the analysis of crystallization statistics, is applied to the calculation of continuous cooling curves for anorthite and o-terphenyl as representative of inorganic and organic systems. In addition to homogeneous nucleation, bulk as well as surface heterogeneous nucleation are considered. The effects of distributions of heterogeneities with-contact angles between 40 and 100 ° as well as overall concentrations of heterogeneities between 103 and 109 cm -3 axe considered. Heterogeneities with contact angles higher than about 100 ° are shown not to have an effect on the critical cooling rate for typical concentrations of heterogeneities. For liquids containing distributions of heterogeneities, the nucleation behavior is dominated by small concentrations of heterogeneities having small contact angles. Theoretical log (Ivr/) versus (T3rAT2r)-1 curves have been constructed for homogeneous nucleation + heterogeneous nucleation with a single type of heterogeneity and for homogeneous nucleation + heterogeneous nucleation with the heterogeneiries distributed with regard to contact angle. In the former case, the curve is composed of two linear portions; and in the latter case, the curve shows pronounced curvature. The curvature reflects a continuous change in the frequency of heterogeneous nucleation. Surface heterogeneous nucleation was assumed to originate at discrete surface heterogeneities and was shown to give rise to continuous cooling curves similar to those calculated for bulk heterogeneous nucleation.

1. Introduction The formal t h e o r y o f t r a n s f o r m a t i o n kinetics [1] has p r o v i d e d the basis for recent approaches to glass f o r m a t i o n where the rate o f cooling o f the liquid is taken as d e t e r m i n i n g w h e t h e r or n o t that liquid will f o r m a glass. The critical cooling rate for glass f o r m a t i o n is thus the rate at w h i c h the liquid m u s t be c o o l e d in o r d e r to avoid detectable crystallization. Since glass f o r m a t i o n requires the avoidance o f detectable crystallization, the kinetic t r e a t m e n t o f glass f o r m a t i o n [2] requires a k n o w l e d g e o f the n u c l e a t i o n 0 022-3093/81/0000-0000/$02.50

© North-Holland

38

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

frequency and the rate of crystal growth over a range of temperature. Data on the crystal growth rates can be obtained experimentally with no considerable difficulty, whereas nucleation frequencies are difficult to measure and are usually calculated using classical nucleation theory. The kinetic treatment of glass formation has been applied to a variety of systems, including silicates (see e.g. refs. [3,4]) and metal alloys (see e.g. refs. [5,6]), to estimate the critical cooling rates. Experimental techniques were developed to determine time-temperature-transformation (TTT) curves for various materials. From these curves, which show the times at various temperatures required to obtain a given volume fraction crystallized in a material [2], nucleation rates can be directly calculated. The nucleation frequencies thus obtained for Na20" 2 SiO~ [4], two lunar compositions [7,8] and anorthite [9] are in reasonable agreement with predictions of the classic theory of nucleation. In all these systems , nucleation has been shown to take place homogeneously under the experimental conditions considered - i.e., undercoolings of the order of 0.2TE--0.35TE (where TE is the melting point or liquidus temperature). More recently, the kinetic treatment of glass formation has been applied [ 10] to interpreting differential thermal analysis (DTA) results for glass-forming systems. It has often been observed that upon slow reheating of a glass in a DTA instrument, an exothermic peak appears at a temperature between the glass transition, Tg, and the melting point, T E. This peak corresponds to the heat of fusion released during the crystallization of the glass. The position of this peak changes with heating rate; and this variation can provide information about the barrier to crystal nucleation. The DTA results in conjunction with detailed crystallization calculations (to be outlined below) for several glass-forming systems have yielded nucleation barriers which are in accord with measurements on other non-metallic systems using droplet techniques [8,11]. Several workers have considered the kinetics of crystal nucleation in Li20" 2 SiO2 (see e.g., refs. [12-16]). Internal nucleation of crystals is observed over a range of large undercoolings. The nucleation behavior apparently does not depend on the crucible used to melt the samples; and the occurrence of homogeneous nu. cleation has been suggested. The kinetics are not, however, well described by the classical theory of such nucleation. Strnad and Douglas [17] studied nucleation and crystallization in three sodalime-silica glasses of silica content between 55 and 60 wt.%. They measured the frequency of bulk as well as surface nucleation for the three compositions. The bulk (internal) nucleation was characterized as homogeneous because of the high degree of supercooling at which the nucleation was observed (AT ~ 0.4TE). The surface nucleation which takes place at heterogeneous sites on the glass surface was observed to peak at a higher temperature than the internal, apparently homogeneous nucleation; and the rate of such surface nucleation was much lower than that of the homogeneous nucleation. A discussion of these and other glass forming systems which exhibit homogeneous nucleation is given by Hinz [ 18].

H. Yinnon, D.R. Uhlrnann / A kinetic treatment o f glass formation V

39

The examples described above, where homogeneous nucleation has apparently been observed, are rather exceptional in glass-forming systems. Such nucleation can be observed by cooling to the range of high undercooling or by reheating after cooling to temperatures below the glass transition. In most cases, nucleation takes place heterogeneously - whether on second-phase material distributed through the bulk, or more often, on surface heterogeneities. A recent review of heterogeneous nucleation can also be found in ref. [18]. Heterogeneous nucleation at glass surfaces has not been studied rigorously, even though it is the most common origin of devitrification. Such nucleation may occur whether the surface of the glass is in contact with a container wall or exposed to atmosphere. In the latter case, it has been suggested that the nucleation is associated with condensed second-phase material, rather than with the existence of the surface per se [ 19] ; and the nucleation sometimes noted at internal bubble surfaces has been suggested to have a similar origin [20]. Other internal heterogeneous nucleation, such as that observed in SiO2 [21], almost certainly originates at secondphase particles. The effects of nucleating heterogeneities on the critical cooling rates for glass formation were explored by Onorato and Uhlmann [22]. Using classical nucleation theory, they constructed T T T and continuous cooling (CT) curves for several glassforming systems as a function of the contact angle between heterogeneities and crystalline phases. From the CT curves, cooling rates necessary to form glasses were estimated. These indicated that heterogeneities with contact angles 0 larger than about 100 ° do not significantly affect the critical cooling rates, while for smaller contact angles, the critical cooling rates increased significantly with decreasing 0. The present work is an extension of the Onorato and Uhlmann study. The effects of depletion of the nucleating heterogeneities are introduced, as well as the effects of distributions of contact angles of the heterogeneities. Also considered are the effects of heterogeneous nucleation associated with surface heterogeneities. In all cases, the critical cooling rates for glass formation are evaluated from detailed calculations of crystallization statistics.

2. Nucleation kinetics and critical cooling rates for glass formation The steady-state rate of homogeneous nucleation can be expressed [23] : IvH° ~ Nv v exp

3

"

(1)

Here Iv is the rate of nucleation per unit volume;N~v is the number of molecules per unit volume; v is the frequency of atomic transport at the nucleus-matrix interface; o is the specific crystal-liquid surface free energy; and AGv is the difference in Gibbs free energy per unit volume between the liquid and crystal phases.

H. Yinnon,D.R. Uhlmann/ A kinetic treatmentof glassformation V

40

Alternatively, the nucleation rate can be expressed: I i f ° ~ Nv v exp

(_0 0205 ~

],

(2)

where the free energy of forming the critical nucleus is BkT*; T* = 0.8TE; Tr = T/TE; ATr = AT/TE; AT is the undercooling and T E is the melting point. For materials where crystallization involves molecular re-orientation or the breaking of directional bonds at the interface, v can be related to the bulk viscosity, 77, with the Stokes-Einstein relation:

u ~ kT/3rragrl,

(3)

where ao is the molecular diameter. The rate of heterogeneous nucleation may be expressed:

IvHEX.~~v2/3 nvA exp(- O'O205------~Bq),

(4)

where n v is the number of active nucleating heterogeneities per unit volume; A is the surface area per heterogeneity; and ~ has been expressed in terms of: (a) the adhesion energy between the heterogeneous phase and the nucleating phase [24]; (b) the difference in thermal expansion between the liquid and the heterogeneity [25,261; (c) the difference in lattice parameter between the heterogeneity and the nucleated crystal [27] ; and in terms of the contact angle, 0, shown in fig. 1 [27]. In terms of the last parameter, ~ may be expressed: = (2 + cos 0)(1 - cos 0)2/4.

(5)

The number of active nucleating heterogeneities per unit volume, nv, does not remain constant during the course of nucleation; rather, the available heterogeneities are depleted as nucleation proceeds, causing a decrease in n v. This depletion should be most marked for small values of 0, and its effect on nv can be approximated: nv(t) = nvo (1 - IvHET t ) , where

(6)

nv(t) is the concentration of nucleating heterogeneities at time t, and nvo is

N

,. H Fig. 1. The contact angle 0 between the nucleus N, heterogeneous phase H, and the supercooled liquid L. (schematic).

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

41

the initial concentration of such heterogeneities. This expression assumes that each heterogeneity provides a single nucleus for crystallization. In the discussion so far it has been assumed that the crystallization time is sufficiently long that a steady-state concentration of subcritical embryos is achieved, producing a nucleation rate which is independent of time [28]. Before such a steady state situation is achieved, the nucleation frequency can be approximated [29,30]. Iv(t) = Ivss e x p ( - r / t ) ,

(7)

where I ss is the steady state nucleation frequency and r is the transient time. The transient time can be expressed, to order-of-magnitude accuracy as: r HOM ~ (n*)2/ns v ,

(8)

where n* is the number of molecules in the critical nucleus, n s is the number of molecules in the surface of the critical nucleus, and v is the frequency of molecular transport at the nucleus-matrix interface, defined in eq. (3). The transient time, r, has also been suggested to be a function of the contact angle between the heterogeneity and the nucleating phase [31 ] as: 7.HET=THOM(1

COS0 ) 2C2--20

.

(9)

Using eq. (9), Gutzow and Toschev were able to calculate the angle 0 for various metals acting as nucleating agents for NaPO3 glass. From eqs. (3) and (8), it is clear that r will increase with increasing viscosity. Table 1 gives the transient times, r H°M, calculated for various temperatures for the two systems studied in the present work. The concepts of the kinetic treatment of glass formation have been described in detail in several papers (see summary in ref. [32]). A solid body is considered to be amorphous if no more than a certain detectable fraction of its volume is crystalline (Vc/IO. The limit of detectable crystallization (by X-ray diffraction, e.g.)has been

Table 1 Calculated transient times for systems studied Tr

0.95 0.90 0.85 0.80 0.75 0.70

Anorthite

o-terphenyl

T (K)

r HOM (s)

T (K)

r HOM (s)

1732 1640 1550 1458 1367 1276

3.10 -3 8.10 --4 9.10 -4 7.10 -3 8.10 -2 6

312 295 279 262 246

3.10 -3 7.10 -3 1.10 -1 5.101 5.10 s

42

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

estimated as V c / V ~ 10-6; but the specific value used has only a minor effect on the calculated critical cooling rates. The last quantity, which represents the minimum rate at which a liquid has to be cooled to form a glass, can be calculated in several ways. The first involves [2] the construction of time-temperature-transformation (TTT) curves, which delineate the times at various temperatures required to form the just-detectable degree of crystallinity. The TTT curves are usually obtained using the relation:

Vc/V=n--[ uat 4 3 v

(10)

where u is the crystal growth rate; Ve/V is the volume fraction crystallized corresponding to a just-detectable degree of crystallinity; and t is the time at each temperature. Each TTT curve has a "nose" which corresponds to the least time at any temperature required to form the given Ve/V , and represents the temperature of maximum overall crystallization rate in isothermal experiments. The critical cooling rate can be estimated from the time and temperature at the "nose" of the TTT curve, using the relation: dT[ ~ TE-- TN - d t -gc ~ tN ,

(I1)

where TE is the melting point, TN is the temperature of the "nose" of the TTT curve, and tN is the time at the "nose". Estimates obtained in this way should overestimate the difficulty of forming glasses, as discussed previously [22]. A more realistic estimate of the critical cooling rate can be obtained by constructing continuous-cooling (CT) curves from the TTT curves which take into account a continuous cooling of the liquid according to a prescribed schedule from T E. It is usually assumed [33] that the cooling takes place at constant rate, although other cooling programs can be treated [22]. A third method of estimating critical cooling rates, which was used in the present work, involves the detailed calculation of the number and size distributions of crystals within a cooling liquid at each temperature, as a function of the cooling rate. In this method, which was introduced by Hopper et al. [34] and used subsequently [10] to simulate crystallization upon reheating a glass, the glass is assumed to cool at a constant rate from the equilibrium temperature, and the number of critical nuclei formed in each temperature (and time) interval is calculated using viscosity data together with assumed or measured values of the nucleation barrier with eqs. (2) or (4). The sizes of the crystals are then evaluated from the times at which they nucleated together with measured crystal growth rates. The volume fraction crystallized in the glass at any time (temperature) is given as the sum over all the nuclei [35]: --~ ( t / ) ~ 47r R.a =i=1 3 t (ti, ti)Ivi(ti) A t ,

(12)

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

43

where VclV(t/) is the volume fraction crystallized up to time tj and Ri(t], ti) is the radius at time tj of nuclei nucleated at time t i. The latter quantity is given as J Ri-R

i* + ~ u k ( t g ) k=i

At

(13)

Here R ; is the size of the critical nucleus at time ti; uk is the crystal growth rate at time interval tk; I~ is the nucleation frequency at time ti and At is the duration of the time interval. A linear cooling rate is assumed, i.e. T = T E - at,

(14)

where T E is the equilibrium temperature and a is the cooling rate. For some assumed cooling rates, time and temperature values exist where V c / V reaches the just-detectable degree of crystallinity, e.g., V c / V = 10 -6. The set of these values comprises a continuous cooling curve obtained without the simplifying assumptions generally used in obtaining such curves. The critical cooling rate is the slowest rate which produces a solid having Vc] V less than the just-detectable degree of crystallinity. This procedure, referred to here as the crystallization statistics analysis, has been carried out for anorthite and for o-terphenyl, for which the viscosity and the crystal growth rate as a function of temperature have previously been determined experimentally. As noted above, the nucleation barriers have also been determined for both of these materials [9,10]. The results are in good agreement with cooling rates estimated using the standard CT analysis, and are in close accord with experience in forming the materials as glasses.

3. Heterogeneous nucleation in the bulk Based on the results of droplet nucleation experiments by several workers, Onorato and Uhlnann [22] inferred a concentration of nucleating heterogeneities in typical liquids as 107 cm -3 or less; and this value was used throughout that work in evaluating the effects of nucleating heterogeneities on glass formation. The present investigation will consider the effects of different concentrations of heterogeneities, ranging from 103 to 109 cm -3. The present work will also take account of the decrease in population of potential heterogeneities as nucleation proceeds by subtracting those which have caused nucleation. This effect is of greatest significance for small contact angles and slow cooling rates, and was not considered in the previous work. The size of the nucleating heterogeneities will again be taken as 500 A (see discussion in ref. [22] for rationale). Specific consideration will be given to o-terphenyl and anorthite as representative of organic and inorganic materials which will form glasses if cooled reasonably rapidly and will crystallize if cooled slowly.

44

H. Yinnon, D.R. Uhlmann / A kinetic treatment of glass formation V 320

I

310

40*

300 29O T(K) 280 270 -

I

i

i

I

~-"

I

I

I

o -Terphenyl

l

0,95

0.90

Tr

90

ooO'°"°°°°°"" 0.85 100 ; " "

I

I

[{~'/* Xucl'e°~te;e°Uly~ 0.80

260 250 -4

I

1700

I

I

-3

-2

I

I

I

-I 0 I 2 Loglo t (minules)

I

I

5

4

~

I

I

I

4~5o~-~~

I

I

0.95

Anorthi 0.90

1600 1500 T(K)

/

I" I/ /

1400 -

- 0.85

/

.....................

l

~

,'.~/ nucleation only

/

I

I

I

-4

-3

-2

I

0.80

'..'~,,- Homogeneous

/

|

1300

1200

Tr

.

0.75 0.70

I

I

-I 0 I Loglo t (rn inures )

I

2_

I

4

Fig. 2. Continuous cooling curves for homogeneous nucleation + bulk heterogeneous nucleation with contact angles as indicated as well as for homogeneous nucleation only: a, o-terphenyl; b. anorthite.

Figure 2 shows continuous cooling curves calculated using the crystallization statistics analysis for o-terphenyl and anorthite with nvo = 107 c m - 3 and contact angles, 0, from 40 ° to 100 °. Similar to the previous conclusion [22], heterogeneities in such concentrations with contact angles greater than about 100 ° do not alter the continuous cooling curve obtained considering only homogeneous nucleation. As shown in fig. 2, even the continuous cooling curve for 0 = 100 ° does not differ greatly from that considering only homogeneous nucleation. Decreasing contact angles beyond this range has a progressively greater effect on the continuous cooling curves, in the direction of making glass formation more difficult.

11. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

45

It can also be seen by comparing the curves in figs. 2a and 2b that the sets of cooling curves for systems with heterogeneities of various contact angles are quite similar in form for anorthite and o-terphenyl. The temperatures and times are, of course, quite different for the two materials. It is appealing to attempt to construct universal sets of curves, appropriate for all materials, by using appropriate reduced times and reduced temperatures. Among such reduced temperatures, consideration has been given to T/TE, T/Tg, T/(TE - Tg), and a form of Hruby's [36] Kgl parameter for describing the temperature of crystallization on reheating a glass, ( T - Tg)/ (T E -7"). Consideration has also been given to scaling parameters for the time based on the viscosities at various fractions of TE. Among the various approaches to constructing "universal" continuous cooling curves, the reduced temperature parameter T - Tg/TE - T provides the best normalization of the results on anorthite and o-terphenyl; and the best reduced time parameter involves scaling the times by the viscosity at temperatures of about 0.87T E. Using these reduced temperatures and reduced times, the results shown in fig. 3 were obtained for o-terphenyl and anorthite. It can be seen that quite close superposition of the CT curves for the two materials is obtained for the cases of homogeneous nucleation and homogeneous nucleation + 107 heterogeneities cm -3 with 0 = 80 °. For the case of heterogeneities with 0 = 40 °, the superposition is not so close, although the "noses" for the two materials are within 0.4 in loglotr. This approach comes surprisingly close to providing universal continuous cooling curves, considering that the viscous flow behavior of simple organic liquids is quite differ-

]

]

[

7-

[

]

I

[

~ . /

6

I

I

Anorthite _

.......... o-Terphenyl

.....'"°°

;5

//

;.': /nvo=lO'tcm-3

4

2

0=40°

ii /

nvo I 0=7 c m ' 3 ~ ~

I

0

I

3

I

4

I

5

I

6

i

I'

-t 8 Loglo( tr )

I

9

I

IO

II

12

Fig. 3. "Universal" cooling curves for anorthite and o-terphenyl for homogeneous nucleation only and for heterogeneous nucleation with e = 40° and 80°.

46

1-1. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

ent from that of oxide liquids, even when use is made of corresponding states parameters [37]. Beyond this, the temperature dependences of the nucleation frequency and the crystal growth rate are different for the different materials. When the range of materials is expanded to include Na20 • 2 SiO2 and a variety of lunar glasses, the temperature at the "nose" of the cooling curve is closely given by: _ rE + 3.5rg

TNOSE

(15)

4.5

COOL

Unfortunately, no simple scaling factor for the time has been found adequate for obtaining universal cooling curves with better than order-of-magnitude accuracy. Figure 4 shows the nucleation frequencies of o-terphenyl as a function of temperature for homogeneous nucleation as well as for heterogeneous nucleation with various contact angles. In all cases, the nucleation frequencies were calculated at the critical cooling rates, and an initial concentration of 107 heterogeneities cm -a was assumed. For 0 = 40 °, 50 ° and 60 °, the abrupt drop in the nucleation frequency is caused by depletion of the nucleating heterogeneities. For 0 = 70 °, the number of heterogeneities drops continuously, but they are not fully depleted during cooling at the critical rate. For 0 > 70 °, there is little or no change in the concentration of heterogeneities during cooling at the critical rate. It is also apparent from fig. 4 that heterogeneities with 0 > 90 ° contribute little to the nucleation rate compared with the rate of homogeneous nucleation, at least for the assumed concentration of heterogeneities. Tr

i

0.70

0.75

0.80

0.85

0.90

I

I

I

I

I

IO I--

8/

I-

0.95 0~

o-Terphenyl

6o*

oo,\

I\ 220

240

260 280 Temperoture ( K )

300

520

Fig. 4. Calculated nucleation frequencies for o-terphenyl for bulk heterogeneous nucleation with contact angles as indicated and for homogeneous nucleation.

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V I

I

I

I

I

I

47

I

Anorthite . n v o = l o g e m "3 7 .E E6 ,,c

_

~ ' ~ ' nvo--107cm-s ~ ' n v o =105cm'3

•~, 5 g4 o __1

3

ql............

Homogeneous nucleation only

.

I

40

I

!

I

I

I

50 60 70 80 90 Contact angle(degrees)

I

IO0

Fig. 5. Calculated critical cooling rates as a function of contact angle for several initial concentrations of bulk heterogeneities. The critical cooling rate for homogeneous nucleation only is shown for comparison.

The effect of different concentrations of heterogeneities on the critical cooling rate of anorthite is shown in fig. 5 for nvo from 103 to 109 cm -3. Also shown for comparison is the critical cooling rate obtained when only homogeneous nucleation is considered. As expected from eq. (4), changes in the concentration of heterogeneities have a smaller effect on the critical cooling rate than changes in the contact angle. At 0 = 70 °, for example, changing nvo by four orders of magnitude from 103 to 107 cm -s changes the critical cooling rate,R c, by slightly less than an order of magnitude, an effect produced by a change in contact angle of about 9 °. The appearance of incipient saturation in log Re shown at small contact angles in fig. 5 is a reflection of the rapid depletion in nucleating heterogeneities found for this range of angles. This effect is seen also in fig. 2, where the continuous cooling curves tend to group together at small contact angles. It is likewise seen from fig. 5 that the contact angle above which heterogeneities have little effect on the process of glass formation increases somewhat with increasing concentration of heterogeneities. Changes in nvo from 103 cm -3 to 10 9 cm -3 change this "critical" contact angle by about 10 °.

4. H e t e r o g e n e o u s nucleation in the bulk with distribution5 o f c o n t a c t angles

The heterogeneities found in a given supercooled liquid are generally not all of the same kind, and may vary in size, shape and contact angle. As noted in the pre-

48

H. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

vious section, the effect of the contact angle has a dominant effect on the nucleation behavior. Hence it seems useful to investigate the effect of a given number of heterogeneities characterized by a distribution of contact angles. In exploring this effect, the distributions of contact angles were assumed to be Gaussian in shape. Several such distributions have been considered, of which two will be discussed here to illustrate the variations involved. Both these distributions have a standard deviation, o, of 10°; one is centered around 0 = 90 ° (henceforth designated as distribution A), and the other is centered around 0 = 60 ° (designated distribution B). In the calculations, the total number of heterogeneities per unit volume (taken as 107 cm -a) was divided into 13 groups representing contact angles ranging from the mean minus 3o to the mean plus 3o in intervals of 5°. On cooling, each group of the distribution nucleates at a different rate according to eq. (4) and thus is depleted at a different rate (at different times). The groups at small contact angles, particularly in distribution B, are low in concentration and are quickly depleted (these heterogeneities are effective at relatively small undercoolings and hence at relatively short times of cooling). With the progressive depletion of the various groups during cooling, the distribution of remaining heterogeneities changes. This is shown in fig. 6 for different portions of distribution B for o-terphenyl cooled at 1000 K min -1 . Since heterogeneities with small contact angles cause nucleation at high temperatures where the crystal growth rate is high, they can dominate the crystallization process even when present in small concentrations. Fig. 7 shows the cooling curve calculated for o-terphenyl containing distribution A of heterogeneities.

il

1

I

i

I

:.4 o g'3

~

I

i

I

t

I

o-Terphenyl

60° 55°

o 0

°l

4.8

/

I

14.4

I

24.0

35.6

43.2

~

I

52.8

Time ( minutes x 103)

Fig. 6. Calculated changes in concentration of the different fractions of distribution B as a function of time as o-terphenyl is cooled at 1000 K rain-1 . Fractions not shown (75° through 90° ) do not change in concentration.

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

300

~290

I

o-Terphen

49

I

...-'''""'-

/

E 280

-.~--- DIstribuhon A' 270

|~

I -I

~ Distribution

I O

A

Logto t (minutes)

I I

Fig. 7. Calculated continuous cooling curves for o-terphenyl. Dashed line, assuming distribution A of heterogeneities; solid line, assuming distribution A minus the two fractions of lowest contact angle (60° and 65°), (Distribution A'). The cooling curves for the two lowest contact angle fraction (60° and 65°) are indistinguishable from the cooling curve for distribution A.

The cooling curves for the two fractions o f the distribution with smallest contact angles (60 ° and 65°), are identical to the curve for the whole distribution, although they represent a very small fraction o f the heterogeneities (3 × 10 -s o f the distribution for the 60 ° fraction and 5 × 10 -4 of the distribution for the 65 ° fraction). Also shown in fig. 7 is the cooling curve for the remaining fractions of the distribution (0 between 70 ° and 120 °) which is considerably different from the cooling curve for the whole distribution. This clearly indicates how a small percentage of the total nucleating heterogeneities having small contact angles can dominate the crystallization process. For distribution B, the situation is complicated by the phenomenon shown in fig. 1 where the cooling curves at small contact angles are close to each other due to depletion of heterogeneities. Thus a change in contact angle between 30 ° and 40 ° does not produce as large a change in the cooling curve as a change between 60 ° and 70 ° . Similarly, the cooling curves for the small contact angle fractions of distribution B tend to overlap; and the crystallization o f the liquid containing such a distribution is not dominated by heterogeneities with a single contact angle, but by heterogeneities in three or four groups of the smallest contact angles. In fact, the cooling curve obtained for heterogeneities corresponding to the 5 lowest contact angle fractions in distribution B (30 ° through 50 °) is identical to the cooling curve obtained when the whole distribution of heterogeneities (30 ° through 90 °) is considered. For perspective, heterogeneities having a contact angle between 30 ° and

50

1-1. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

50 ° constitute only about 13% of the total number in distibution B. The changes in the rate of nucleation with time and temperature, for various distributions of heterogeneities are reflected in a plot of log Iv~ versus (Tr3AT2)-1 . The combined rates of homogeneous + heterogeneous nucleation were calculated for a variety of distributions of heterogeneous nuclei. Typical results for o-terphenyl are shown in fig. 8 for: (1) distribution A of heterogeneities present in a total concentration of 107 cm-3; (2) a concentration of 106 heterogeneities cm -3, all with 0 = 80°; and (3) only homogeneous nucleation. The curves for cases (1) and (2) include contributions from both homogeneous and heterogeneous nucleation; and in all cases, the calculations involved holding the samples isothermally for five minutes at the indicated temperatures (to simulate typical nucleation experiments). When depletion of some groups in the heterogeneity distribution caused a time dependence of the nucleation rate, the indicated nucleation frequency reflects the average value over the period of 5 rain. For the single type heterogeneity with 0 = 80 °, the log Ivr~ versus ( T r3A T r2) -1 plot is composed of two straight lines of different slope with some curvature near their intersection. It is apparent (compare the 0 = 80 ° curve with the homogeneous nucleation only curve) that at low temperatures, the nucleation is largely homogeneous, while at high temperature heterogeneous nucleation dominates. According

18

I

16

I

I

I

I

\

"7 14

I

I

I

o -Terphenyl

-

~1~

12

~ o..

~ X%

DistributionA (nvot0 cm-3)

+

g 6

\". ~,,,,~.~ .~,~~.,,

_1 4

~. +oo

nucleotion only

0 0

I 20

I 40

~ : I~ 60

~

~'~ ~

I 80

_

o.o"---._

~ - v.v ~ ~,.~ \ nvo--IL~ cm'~~ I~ I I I I00 120 140 160 180

(Tr3A TrZ)-' Fig. 8. log 10(Ivr/) versus (TraATr2) -1 for homogeneous nucleation only (dotted line), homogeneous + heterogeneous nucleation with a single type of heterogeneity (0 = 80 °) (solid line), and homogeneous + heterogeneous nucleation with distribution A of heterogeneities (60 ° < 0 < 120°).

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

51

2 -1 to eq. (4), the slope of the log Ivr/versus (Tr3 ATr) plot reflects the barrier to crystal nucleation, and thus should depend on the contact angle when heterogeneous nucleation is considered. In the case of heterogeneities with a distribution of contact angles (the more generally expected case), the nucleation frequency is dominated at each temperature by a different type of heterogeneity. At high temperatures (small undercoolings), only heterogeneities with small contact angles contribute to nucleation; hence the slope at such temperatures is small. As the temperature decreases, heterogeneities with progressively increasing contact angle dominate the nucleation process, leading to progressively larger slopes on log Iv~ versus (Tar/XT2r)-1. The result, as shown in fig. 8, is a continuous change in slope until at sufficiently low temperatures, homogeneous nucleation begins to dominate and the slope remains constant. From these results, it is apparent that experiments where the nucleation frequency is measured over a reasonable range of large undercoolings should permit homogeneous nucleation to be distinguished from nucleation dominated by hetero2 -1 geneities: (a) from the magnitude of the intercept on the log Iv77versus (T r3 AT r) plot, which gives the pre-exponential constant; (b) from the magnitude of the slope, which gives the nucleation barrier; and (c) from the presence or absence of curvature in the plot.

6. Nucleation at surfaces

Heterogeneous nucleation associated with the external surfaces of a body can be treated in a similar manner to the treatment of heterogeneous nucleation in the bulk, but with the heterogeneities found only on the surfaces of the cooling liquid. For the present calculations, the heterogeneities were assumed to be 500 A spheres (similar to the case of heterogeneous nucleation in the bulk) half embedded in the liquid so that only half of their surface area is available for nucleation. Experience with crystal growth originating at surfaces of many glasses has indicated a mean distance between nucleating heterogeneities of the order of 2 0 - 3 0 #m. In the present analysis, a mean distance of 30/am was assumed, which corresponds to a heterogeneous phase aerial fraction of approximately 2.5 × 10 -6. Since all crystallization is taken as initiating only on the surface, the criterion of a volume fraction crystallized of 10 -6 as the limit of detectable crystaUinity can be misleading. Crystallization could be detected even with the naked eye before VcJV reaches a value of 10 -6 depending on the dimensions of the sample studied. Thus the criterion of just-detectable crystallization was changed to 1% of the surface area. For a cube 1 cm on an edge, this surface area criterion is equivalent to the volume fraction criterion of 10 -6. For smaller sample dimensions, the volume fraction crystallized of 10 -6 is reached sooner in the cooling process, before the aerial fraction of 10 -2 is obtained. The critical cooling rate, Re, for just-avoiding an aerial fraction crystallized of

52

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

10 -2 depends only on the number of nucleating heterogeneities per unit area of surface and on the contact angle 0. The dependences ofR c on both of these parameters are similar to those found for the case of bulk nucleation (as depicted in fig. 5). Two distinct stages can be identified in the growth of the surface nuclei. In the first, the nuclei are relatively widely spaced and grow independently of one another. In this stage, the volume fraction crystallized is calculated by summing the contributions from the individual nuclei, as was done for heterogeneous and homogeneous nucleation in the bulk. When the nuclei grow to the extent that their combined cross-sectional area equals the total area of the surface, the nuclei are assumed to form a uniform layer growing inward. In this second stage, the volume fraction crystallized, Vc/V, is given by:

V--£c 1 = V1 - (

---2Utc)3 '

(16)

where u is the crystal growth rate, t c is the time of growth after consolidation of the growth fronts from the individual nuclei, and s is the linear dimension of the sample. Taking a just-detectable degree of crystaUinity as an aerial fraction crystallized of 10 -2, only first-stage growth affects the critical cooling rates. In considering further growth originating at surface heterogeneities, however, both stage 1 and stage 2 growth must be considered. Fig. 9 shows the increase in the volume fraction crystal-

I

-2

0-Terphenyl

I

I

:;~ - 8 v o o

STAGE 2

_J -12 -14 -16 -18

508

I

296

284 Temperature ( K )

272

260

Fig. 9. Calculated crystallization of a 1 cm 3 cube of o-terphenyl assuming both homogeneous and heterogeneous nucleation with 0 = 70 ° . The t w o crystal growth stages are shown (see text).

1-1. Yinnon, D.R. Uhlmann / A kinetic treatment o f glass formation V

53

lized with time for a 1 cm a cube of o-terphenyl. Second-stage growth begins at an aerial fraction crystallized of 1 and a volume fraction crystallized of approximately 0.01. There is a smooth transition between the two stages of growth.

7. Discussion

The approach of crystallization statistics as used in the present work does not take into account any time dependence of the nucleation rate associated with the time required to establish a steady-state distribution of subcritical embryos. To study the possible effects of a time-dependent nucleation frequency on the critical cooling rate, the crystallization statistics routine was modified to include an exponential build-up of the nucleation frequency as indicated by eq. (7). The transient times were calculated using eq. (8). At each temperature interval during the cooling process, a new transient time was calculated; and the nucleation frequency allowed to change according to eq. (7). The overall number of nuclei formed during each time interval was calculated by summing the variable nucleation frequency over the time interval considered. In the case of homogeneous nucleation, the critical cooling rates for both anorthite and o-terphenyl are sufficiently low that the transient time, r, is considerably smaller than a typical time interval (the time that the system takesto cool a temperature interval equal to 2% of the temperature range between TE and Tg). Hence steady-state conditions should maintain, and the nucleation rate can be considered as independent of time at each temperature. When heterogeneous nucleation is introduced, and faster cooling rates are required, the value of r can become close to the typical time interval; and thus the nucleation rate at high undercoolings can become significantly dependent on time. In such cases, however, the crystallization process is dominated by heterogeneous nucleation which takes place at temperatures closer to the melting point, where ~is considerably smaller; and thus the critical cooling rate with heterogeneous nucleation is affected only to a very small extent by transients. Calculations have shown that only at the fastest critical cooling rates, typical of heterogeneities with contact angles of 40 ° or 50 °, is there a decrease in the nucleation frequencies due to the time dependence of Iv; and even in these cases, the decrease is small. The change in the critical cooling rate as a result of the time effect is very small, suggesting that in light of the approximations implicit in eqs. (7) and (8), transient nucleation can safely be assumed as of small significance in the present treatment. The results of the present study have confirmed the original suggestions of Onorato and Uhlmann [22] concerning the effects of nucleating heterogeneities on glass formation. The present work used the analysis of crystallization statistics in treating crystallization during continuous cooling, while the previous authors used the more approximate cooling analysis. It was found that heterogeneities with con-

54

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

tact angles greater than about 9 0 - 1 1 0 ° will have a negligible effect on the critical cooling rate for glass formation, at least for concentrations of heterogeneities in the range of 10 a to 109 cm -3. Specific account has been taken of the depletion of nucleating heterogeneities, which reduces the effect of changes in contact angle on critical cooling rate. It has been shown that modest changes in contact angle have a comparable effect to large changes in the concentration of heterogeneities. A small concentration of heterogeneities with small contact angle can dominate the crystallization process on cooling a liquid. Such heterogeneities cause nucleation at relatively high temperatures where the growth rate is high, and hence the rate of overall crystallization is high when the nuclei are present. This effect was demonstrated by calculations for various distributions of heterogeneities, where the nucleation process was dominated by the group(s) of lowest contact angle. This conclusion should be recalled when performing controlled heterogeneous nucleation experiments. Liquids containing different (known)concentrations of certain types of nucleating heterogeneities (e.g., platinum powder) can nucleate at the same rate due to the presence of a small concentration of heterogeneities with a lower contact angle. Furthermore, since the concentration of such heterogeneities with small contact angles will usually vary from one sample to another, apparently erratic nucleation behavior can result. The presence of unsuspected heterogeneities can usually be indicated by the plot 3 2 -1 of log Ivr~ versus (TrATr) . As shown in fig. 8, deviations from straight lines in such plots can indicate the presence of nucleating heterogeneities. Pronounced curvature would suggest that heterogeneities are of several different types with different contact angles. Finally, the case of surface nucleation has been shown to be similar to the case of bulk nucleation, deviating only in the criterion for defining a justdetectable degree of crystallinity. 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 and Mr. D. Cranmer of MIT.

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[8] D.R. Uhlmann, C.A. Handwerker, P.I.K. Onorato, R. Salomaa and D. Goncz, Proc. Ninth Lunar and Planetary Science Conf. (Pergamon, New York, 1978) p. 1527. [9] D. Cranmer, R. Salomaa, H. Yinnon and D.R. Uhlmann, J. Non-Crystalline Solids, to be published. [10] P.I.K. Onorato, D.R. Uhlmann and R.W. Hopper, J. Non-Crystalline Solids 41 (1980) 189. [ 11 ] H. Yinnon and D.R. Uhlmann, to be published. [ 12] K. Matusita and M. Tashiro, J. Non-Crystalline Solids 11 (1973) 471. [13] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [14] E.G. Rowlands and P.F. James, Phys. Chem. Glasses 20 (1979) 1. [15] E.G. Rowlands and P.F. James, Phys. Chem. Glasses 20 (1979) 9. [16] G.F. Neilson and M.C. Weinberg, J. Non-Crystalline Solids 34 (1979) 137. [17] Z. Strnad and R.W. Douglas, Phys. Chem. Glasses 14 (1973) 33. [18] W. Hinz, J. Non-Crystalline Solids 25 (1977) 216. [19] D.R. Uhlmann, J. Non-Crystalline Solids 41 (1980) 347. [20] J.E. Neely and F.M. Ernsberger, J. Am. Ceram. Soc. 49 (1966) 396. [21] F.E. Wagstaff, J. Am. Ceram. Soc. 52 (1969) 650. [22] P.I.K. Onorato and D.R. Uhlmann, J. Non-Crystalline Solids 22 (1976) 367. [23] D. Turnbull and M.H. Cohen, in: Modern aspects of the vitreous state, Vol. 1, ed., J.D. Mackenzie (Butterworths, London, 1961). [24] 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) p. 10. [25] I. Gutzow, Glastech. Bet. 46 (1973) 219. [26] S. Toschev and I. Gutzow, Phys. Stat. Sol. 24 (1967) 349. [27] D. Turnbull, in: Solid state physics, Vol. 3 (Academic, New York, 1956). [28] R. Becket and W. D6ring, Ann. Phys. 24 (1935) 719. [29] J.B. Zeldovich, Aeta Physiochim. URSS 18 (1943) 1. [30] D. Turnbull, Trans. Am. Inst. Mining Met. Engrs. 175 (1948) 774. [31] S. Toschev and I. Gutzow, Phys. Stat. Sol. 21 (1967) 683. [32] D.R. Uhlmann, J. Non-Crystalline Solids 25 (1977) 42. [33] R.A. Grange and J.M. Kiefer, Trans. ASM 29 (1941) 85. [34] R.W. Hopper, G. Scherer and D.R. Uhlmann, J. Non-Crystalline Solids 15 (1974) 45. [35 ] H. Yinnon, A. Roshko and D.R. Uhlmann, Proc. 1 l th Lunar and Planetary Science Conf. (Pergamon, New York, 1980) p. 197. [36] A. Hruby, Czech. J. Phys. B22 (1972) 1187. [37] W.T. Laughlin and D.R. Uhlmann, J. Phys. Chem. 76 (1972) 2317.