Crystallization statistics, thermal history and glass formation

Journal of Non-Crystalline Solids, 15 (1974) 45-62. © North-Holland Publishing Company

CRYSTALLIZATION

STATISTICS, THERMAL HISTORY

AND GLASS FORMATION R.W. HOPPER, G. SCHERER and D.R. UHLMANN Department of Metallurgy and Materials Science, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Mass., USA Received 8 August 1973 Revised manuscript received 28 September 1973

The formal theory of transformation kinetics describes the volume fraction of a phase transformed in a given time at a given temperature. The basic concepts are extended for isotropic crystal growth in a material having a known thermal history T(r, t). A crystal distribution function ~(r, t, R) is defined such that the number of crystallites in a volume do at r having radii between R and R + dR at time t is qJ(r, t, R) do dR. The function q; contains essentially complete statistical information about the state of crystallinity of a material. Formal expressions for ~ are obtained. Applications are discussed, including predictions of crystallinity when T(r, t) is known; predictions of glass-forming tendencies; experimental determination of nucleation rates; and the determination of the thermal history of a sample from post mortem crystallinity measurements. As an example, qJ(r, t, R) is calculated for a lunar glass composition subjected to a typical laboratory heat treatment.

1. Introduction The formal theory of transformation kinetics was developed by Johnson and Mehl [1 ] and Avrami [ 2 - 4 ] to describe the fraction of a phase transformed in a given time at a given temperature. These theoretical treatments relate the volume fraction transformed F v to the frequency of nucleation of the new phase (13 and to its growth rate (u). In general, ! and u can be functions of both temperature and time; and for various phase morphologies and time dependences, expressions are available for F v versus time at given temperatures. The most frequent use of such analyses has involved the construction of 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 ( T - T - T ) curves corresponding to a given fraction transformed. These are constructed by selecting a particular fraction transformed, calculating the time required for that volume fraction to form at a particular temperature, and repeating the calculations for other temperatures and other fractions transformed. In a recent extension of this analysis, Uhlmann [5] has considered the kinetic conditions for glass formation by constructing T - T - T curves corresponding to a

46

R. 14,'.Hopper et aL, Crystallization statistics, thermal history and glass formation

just-detectable degree of crystallinity (F v -- 10-6). The curves were generally constructed from measured growth rate versus temperature data and from calculated nucleation frequencies. The latter were calculated from measured viscosity data using the standard model for homogeneous nucleation with representative values for the crystal-liquid interracial energy. With this approach, the minimum conditions for forming glasses of various materials were estimated. Such estimates included the minimum cooling rates required for glass formation and the maximum thickness obtainable as glasses. It was concluded from the analysis 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. While this analysis provided useful insight into the glass-forming abilities of different materials using both nucleation and crystal growth rates (rather than adopting a criterion of glass formation based only on nucleation processes), it provided no information about the size and number distributions of crystals in a nominally glassy or partly devitrified body subjected to a given thermal history. In many applications, however, such detailed information is important; and in several notable cases - such as in elucidating the thermal histories of various lunar samples - the detailed distributions and their relation to the kinetic characteristics of the materials are essential for obtaining the desired information. In the present paper, expressions will be obtained for the distributions of the crystallites which form within a macroscopic body subject to a general thermal history T(r, t). The treatment is predictive in giving distributions of crystal sizes, and is readily amenable to various inversions of the problem such as deducing the thermal history of a sample from measurements of the crystallite distributions.

2. Analysis The analysis of the present section is based on the following assumptions, which are appropriate for many glass-forming systems, and several of which are relaxed in the appendices to follow. The assumptions are: (1) The crystal growth velocity is a function of temperature; i.e., u = u(T). This is appropriate for the interface-controlled growth often encountered in glass-forming systems. Although the present analysis does not consider diffusion-controlled growth, it can directly be extended to that case. Such an extension is discussed in appendix A. (2) Nucleation at the external surfaces of the specimen is ignored, since the analysis is directed to crystallization in the interior of the material. Again, it can readily be extended to include such nucleation. (3) The nucleation rate per unit volume in the bulk of the material (which combines both homogeneous and heterogeneous nucleation) depends only on the instantarious temperature, I v = Iv(T). Thus delay times in nucleation are ignored in the basic analysis. Such delay times are included in appendix B, however, where the

R. I4I. Hopper et al., Crystallization statistics, thermal history and glass formation

47

analysis is extended to the case where the nucleation frequency depends explicitly on time. (4) The body is homogeneous on a macroscopic scale, which in particular is taken to mean that nucleation sites are uniformly distributed throughout the volume of the material. (5) The growth rate is isotropic in an isothermal environment, and the thermal gradients are small on a scale compared with the crystallite size. The growth velocity in the sample will then be independent of direction, resulting in spherical crystallites. Internal nucleation in glass-forming liquids often leads to a spherulitic morphology, and the present analysis is applicable to such growth as well as to other cases where the growth rate anisotropy is small. (6) The volume fraction crystallized in any given region of interest is sufficiently small that the excluded volume problem can be ignored. This assumption is relaxed in appendix C. (17) The temperature problem can be decoupled from the crystal growth problem in that heating due to the latent heat of crystallization is neglected. This may be the most restrictive of the assumptions, for the authors have recently suggested [6] that local heating at the interface of internally nucleated crystals in many glassforming systems can be substantial. Such interface heating could affect the growth rate, u(T). Proper treatment of this effect would require the solution of a coupled temperature problem of formidable proportions. In many cases, however, the effect of interface heating on growth rate can well be ignored. (8) The growth velocity is positive at all undercoolings: u(T) > 0, for T < TE, where T E is the equilibrium melting point or liquidus temperature. This is in accord with theory. With these assumptions, let us define a 'crystal distribution function' ~(r, t, R), such that the number dn of crystals in the volume dv at r having a radius between R and R + dR at time t is given by (1)

dn = ~(r, t, R ) do d R .

This function contains all of the basic statistical information about the crystallinity of the system. A crystal nucleating at the time t o < t has a radius at time t of t

R(r, t, t o) = ( u IT(r, t')] dt',

R> 0

~2)

to

Here the temperature T(r, t) is a specified function of position and time. Note that -the temperature at any position need not vary monotonically with time, and even oscillations through T E are permitted. The only restriction on eq. (2) is that R be positive; that is, if melting leads to the complete disappearance of a crystallite, eq. (2) is no longer applied, since the crystallite would have to be nucleated a second time before growth could again proceed. The important feature of eq. (2), which is per-

R. W. Hopper et al., Crystallization statistics, thermal history and glass formation

48

haps as obvious from the physics as from the equation, is that for a given u(T) and T(r, t), there exists a one-to-one correspondence between R(r, t, to) and t 0. Thus, a crystal having radius R at position r and time t must have nucleated at a unique time t o < t. Eq. (2) therefore also serves to define implicitly a function to(r, t, R), which for fixed (r, t) is single valued in R. The number dn of crystals in do nucleating between t o and t o + dt 0 is, by definition of the volume nucleation frequency Iv(T),

dn = I v [Z(r, to) ] dvdt 0 .

(3)

Note that Iv(T) = 0 for T > T E, i.e., when u(T) < 0. To obtain 4, we require the number of crystals having a radius between R and R + dR at the time t > t 0. This is most easily deduced formally* by noting that the number of crystals nucleating during the finite time interval between t 1 and t 2 (with t 1 < t 2 < t) in the volume element do is

dN12 = do

?

/v [T(r, to) ] dt 0 .

(4)

tl From eq. (2), one obtains OR Ot0

- -u

[T(r,

to)].

(5)

Changing variables in eq. (4) with the aid of eq. (5), letting R 1 = R(r, t, t2) and

R 2 =-R(r, t, t l ) , and noting t h a t R 2 > R 1 , one obtains:

dvf2j Iv{T[r, to(r, t, R)]} dN12

R,

u{T[r, to(r, t , R ) ] } dR"

(6)

Direct comparison with eq. (1) now shows that the integrand of eq. (6) is just

t, R): I v (fir, t0(r, t, R)] f(r, t, R) = . rIr, t o (r, t, m ] )"

(7)

Other statistical distribution functions describing the state of crystallization of a sample can be either derived from ~k(r, t, R) or obtained directly. First note that the largest crystal radius at (r, t) is given by * A more intuitive approach involves breaking up the range of R into equal intervals dR; obtaining the corresponding dto'S from eq. (5), which merely states that crystals nucleating at t o grow an amount dR between time t o and t o +dto; and obtaining the number of crystals nucleating in this time interval from eq. (3).

R. W. Hopper et aL, Crystallization statistics, thermal history and glass formation

49

t Rma x (r, t) = f u IT(r, t')] dt' . 0

(8)

Then the number density of crystals at (r, t) is R max D N (r, t) = f 0

$(r, t, R) dR

(Oa)

t

=f

(Oh)

I v [T(r, to) ] dt o .

o The volume fraction crystallized at (r, t) is R max

F (r, t) = f

~ R 3 ~(r, t, R) dR

(10a)

0 t

%rR(r, t, to)3 lv[T(r, t0)]dt 0 .

(lOb)

0 The average number density of crystals in the sample at time t is a simple volume average,

~N(t) = ~1 f ON(r, t)

d3r.

(11)

V

Likewise, the volume fraction crystallized in the entire sample at time t is 1

f l y ( t ) = ~ f F v ( r , t) d3r

(12)

V

For many purposes, the statistics are desired at a time long after crystallization has ceased (because the sample is cold). This case can be treated with the above analysis by using t cs, the time at which crystallization essentially stopped, as one of the limits on the integrals over time, and the corresponding radius values as the limits on the integrals over R. All the parameters in the analysis are the usual unreduced quantities expressed in a standard, consistent set of units. In many practical applications, however, it is helpful to employ reduced dimensionless parameters, at least in a portion of the analysis. This is particularly appropriate when numerical analysis is used in solving parts of the problem, such as in determining the temperature distribution. In cases where the thermal problem includes a characteristic distance a 0 and a characteristic

50

R. W. Hopper et al., Crystallization statistics, thermal history and glass formation

(constant) thermal diffusivity r 0 the following reduced parameters are helpful: r* = r/ao, t* = Kot/a 2, R* = R/a O, u*( T) = u( T)ao/~ O, I*(T) = Iv(T) a5 /~O ' and 4 " = ~a 4. These are dimensionless; and the equations derived above retain their form unchanged when expressed in terms of the new variables.

3. Discussion

The analysis presented in sect. 2 is based on the same principles as those used in the construction of time-temperature-transformation curves. In comparison with such curves of volume fraction crystal-liquid versus time at a given temperature, the present crystal distribution function if(r, t, R) contains more detailed information about the state of crystallization in any portion of a body (including crystal size as well as volume fraction crystallized), and is based upon specific consideration of the thermal history of different portions of a specimen. The present analysis can be manipulated in various ways to obtain information about a specimen or a material. When u(T), Iv(T) and T(r, t) are known, a prediction of ~(r, t, R) is the most frequent objective. In sect. 4, an example is presented in which ~(r, t, R) is obtained for a lunar glass quenched by pouring onto a thick block of stainless steel. This corresponds to a commonly used qualitative test of the glassforming tendency of an unknown material, and it is useful to enquire more closely into the significance of the test. Where isothermal-transformation ( T - T - T ) diagrams are desired, they can be obtained directly from T(r, t) and Fv(r, t). In addition to these direct calculations, various inversions are possible. Depending upon the assumptions made in the inversion analysis (e.g., an assumption that T(r, t) decreases monotonically in time is often helpful) non-uniqueness in the results may sometimes occur. Usually, however, this can be avoided if full use is made of the available information as a function of R. As an example, if Iv(T) and u(T) are known or can be calculated, and the final crystal density function is measured, then the temperature T O at which a crystal of radius R nucleated is immediately known. In this case, ambiguity exists only iflv/U has the same value for several temperatures. The more interesting example of determining the thermal history from post mortem crystallite statistics will be considered in detail in a later publication [7]. In that work, the present analysis is applied to the case of a semi-transparent glassy body cooling by radiation into a vacuum. The kinetics involved in this problem are important in a number of lunar science studies. In these cases, the complete specification of the number of crystallites and their size distribution can be determined as a function of position by laboratory examination of lunar samples. Particularly interesting in this regard are samples, such as 60095, which are nearly spherical in shape and largely (but not completely) glassy. The isothermal nucleation and growth kinetics can be determined by laboratory experiments; and the combination of such data with the results of the present analysis permits much information to be deduced about the thermal histories of the lunar samples.

R. W. Hopper et aL, Crystallization statistics, thermal history and glass formation

51

The basic approach in determining thermal histories involves measuring u(T) and Iv(T ) in a series of laboratory experiments; then ~cs(r, R), the final crystal distribution function, is measured in a post mortem analysis of the specimen of unknown thermal history. Then in any small region at r, the temperature of nucleation T O of the crystals of radius R is known, because Iv/u is known as a function of T. Thus, for each value of R at r, T[r, t~s (R)] is known. The quantity tgs is determined implicitly from the relation t cs

RCS(r, t~ s) = f

u[T(r, t ' ) l d t ' ,

(13)

cs

to

where R cs is the size of the crystals observed in the specimen (after crystallization has effectively stopped). In practice, it is often convenient to assume a reasonable time dependence for T(r, t) and select the parameters - which in general depend on r - such that eq. (13) produces a best fil to the observations, lterative techniques can also be used to good advantage. Another application of the analysis involves the determination of u(T) experimentally, performing a careful devitrification experiment in which T(r, t) is known, and measuring the final crystal density function, $cs (r, R). In this case, R cs (r, to) is readily determined, and I v ( T ) can be immediately obtained from $cs (r, R). This appears to be one of the more straightforward methods of determining Iv(T ) experimentally. A further application of the analysis is the evaluation of the relative importance of various distributions of nucleating heterogeneities in affecting glass formation. Such an evaulation can be carried out for different viscosity-temperature relations and for various thermal histories, and should indicate the conditions under which different types of heterogeneities are critical to the formation of glasses. It should be noted that neither the analysis in sect. 2 nor the manipulations involved in the above applications involve explicit use of the position vector r. The vector r merely serves as a label for the volume element under consideration, and the statistical treatment must be applied to each element separately. One consequence of this is that the thermal history of a small specimen which was originally part of a larger body can be determined independently of the availability of the original body.

4. A numerical

example

In this section we consider an approximate treatment of a simple problem, that in which an opaque glass is quenched from the melt by pouring onto a thick block of cool solid (steel). This is often used as a rough test of the glass-forming tendency of a material. For simplicity, the thermal problem will be approximated by that for an infinite composite solid. The present analysis will be valid until such time as the

52

R. W. Hopper et al., Crystallization statistics, thermal history and glass formation

remote surfaces of the glass and the steel begin to deviate significantly from their original temperature; one must also exclude the layer of the glass at the free surface, which would cool due to radiation and contact with the air. The authors have also treated the case of a finite glass plate quenched between two metal plates, but the complexity of this thermal problem is unnecessary for present purposes. The thermal properties of stainless steel will be used for the quenching block. The thermal and kinetic parameters employed for the glass will be those of Lunar Composition 14259, which have been described in detail elsewhere [8, 9]. The material has the composition, in wt%: SIO2(48 ), A1203(18), FeO(10), MgO(9.2), CaO(11), TIO2(1.8). It has a liquidus temperature TL of approximately 1240°C. The high iron content makes the material black and quite opaque, although some transparency is noted through sections thinner than about 1 ram. Taking an absorption coefficient of 20 cm- 1, then at the maximum temperatures of interest (perhaps 100°C above TL), the radiation conductivity is approximately equal to the phonon conductivity, each being approximately 0.004 cal. cm -1 • s-1 . Thus, during most of the cooling, the radiation conductivity will be small and the total conductivity approximately constant. The assumption of opacity is therefore not unreasonable for this glass in the temperature range of interest. The crystal growth velocity of this material has been measured [9] over a range between 10 -2.4 cm • min- 1, the maximum observed rate, and 10 -4-5 cm • min- 1. The rates were found to be independent of time at all temperatures. Fifteen of the measured growth rates were used in a computer subroutine which gives u(T) using a linear interpolation scheme. In this and similar materials, it was observed [9] that internally nucleated crystals usually exhibit a spherulitic morphology, so that the morphological assumption of the analysis seems appropriate. The volume nucleation rate has not been measured for this material; we shall therefore include only the homogeneous nucleation rate, which for this material may be approximated:

1.024 T[ Iv(T)= 1034 exp ( - T3-~LL - T)2]"

(14)

This expression, representing homogeneous nucleation, should provide a lower bound to the actual nucleation rate. The temperature in the glass (as a function of distance x into the glass) is given by [10]:

Tg (x, t) = T O + (T O - TsO)

(ggPgCpg)l/2

s

(ggOgCpg)l/2 + (~sPsCps)l/2

X [1

(gsPsCps) 1/2

(15)

R. W. Hopper et aL, Crystallization statistics, thermal history and glass formation

I

1

I

I

53

I

1600

1400 o

1200 o,.

E

iooo x = 0 1. 0 800 L 0

I

4

i

I

8

i

I

12

l

I

16

i

I

20

J 24

Time (seconds)

Fig. 1. Temperature T as a function of t, 0 ~ t <--20 s, at x = 0.10 cm and x = 0.20 cm. Here K is the thermal diffusivity, P is the density, Cp is the specific heat, T O is the initial temperature, and the subscripts g and s refer to glass and steel respectively. The parameters used were ( p C ) = ( p C ) = 1.0 cal • cm - 3 Kg = 0.004 c a l ' c m - 2 " s - 1 and Ks = 0.06 c a l - 1 . c m - 2 . P-J~. p s Using eqs. (2) and (7), the crystal density function, 4(x, t, R ) was calculated for several values of t ranging from 20 sec to 1 h in the interval 0 ~< x <~ 1.0 cm with T° - - 1600 ](and T O= 300 I(. It was found that for this material with the indicated thermal history, there is a rather sharp transition from almost no crystallinity near x-- 0.3 cm. Values of 4 for small x tend to be uninterestingly small and somewhat inaccurate. Hence for purposes o f this example, we shall limit our attention to the interval 0.10 cm ~< x ~< 0.20 cm. On this interval, accurate values o f 4 were computed, and the results are such that there is a sensible degree of crystallinity but not so much as to compromise assumption (6) in sect. 2. Further, over this region o f x and t, eq. (15) represents an excellent approximation to the temperature distribution in a quenched sample 1.0 thick. The thermal history at the limits x = 0.10 cm a n d x = 0.20 cm as computed from eq. (15) is shown in fig. 1. The volume fraction F v computed from eq. (10b) is given in fig. 2. As indicated by the shaded region in fig. 2, the precision o f f v at small values o f f v is low in this calculation. The crystal distribution function 4 is shown on a logarithmic scale in fig. 3 as a function of R. On this scale, 4 ( 0 . 1 0 cm, 20 s , R ) does not appear to differ much from 4 ( 0 . 2 0 cm, 20 s,R). A t R = 50 A, however, there is a factor o f approximately 2.5 difference between the values of 4 at the two distances into the glass. Fig. 4 shows 4 as a function o f x f o r R = 45 A and R = 50 A. A complete set o f curves o f the type presented in figs. 3 and 4, or the large

54

R. W. Hopper et aL, Crystallization statistics, thermal history and glass formation

20

I

I

I

I

I

0.12

0.14

0.16

0.18

0.20

16

'KI"

12

o x

u2 8

4

0 0.10

I

0.22

x (cm) Fig. 2. Volume fraction transformed, F v, as a f u n c t i o n o f x , 0 . 1 0 c m ~< x ~ 0 . 2 0 cm, at t = 20 s.

amounts of tabulated data from which they were computed, constitute essentially complete information about the state of crystallinity of the material. In using such information, it is often helpful to reduce the data to forms having immediate experimental significance. In doing this, let us define a function, N, the number of

26

I

I

I

I

I

24 ~/px = 0.20 22 =

.

o_

14

12 I0

,

0

I

200

i

I

400

i

I

600

,

I

800

J~'~

i

I000

1200

R(1) Fig. 3. L o g l o 0 as a f u n c t i o n o f R; 10 A ~ R ,~ 1 0 0 0 A , at x = 0 . 1 0 cm, and x = 0 . 2 0 cm, and t = 20 s. ~ has units of c m - 4 .

R.W. Hopper et al., Crystallization statistics, thermal history and glass formation

21.2

I

I

I

I

I

21.0 E 20.8

~

20.6

~ 20.4 20.2

55

o

~

50 A

I 0.12

20.0 0,10

I 0.14

I 036 x(cm)

I 0.18

I 0.20

0.22

Fig. 4. L o g l o 9 as a f u n c t i o n o f x, 0 . 1 0 c m ~< x ~< 0 . 2 0 cm, at R = 4 5 A and R = 5 0 A , and t = . 2 0 s. ~ has units o f crn- 4 .

crystals having radii between R - ~ R and R+ ~1 R in a cubic volume element, 10 #m on a side. This number is given by

N=R~ X (10 -10 cm3).

(16)

Fig. 5 s h o w s N as a function o f R forx = 0.20 cm. It is seen that for this material with the indicated thermal history, such a volume element will rarely contain any crystallites larger than about 1000 A in diameter, but will contain large numbers of small crystallites (approximately 106 having diameters between 36 and 44 A, e.g.). A function which is closely related to an important type of laboratory experiment is the number, M, of crystallites having radii between R - ~ R and R + ~ R in a volume elememt 50R X 50R X 4R. This number is: M=R4t~

X 103 .

(17)

S u c h a q u a n t i t y s h o u l d c o r r e s p o n d t o a r e a s o n a b l e t h i n s e c t i o n u s e d in t r a n s m i s s i o n

I

I

i

I

I

I

i

I

i

I

i

6

z

2

Q

-~o -2

-~

0 20

i

0

I

2oo

i

I

i

4oo

I

~00

i

8oo

,000

R(I) Fig. 5. L o g l o N ( e q . ( 1 6 ) ) as a f u n c t i o n o f R , 10 A ~
56

R. 14/.Hopper et aL, Crystallization statistics, thermal history and glass formation

O,

I

I

I

I

I

I

i

I

I

I

i

I

I

I

i

N o!'-×=0.10~ -7

I

I

200

i

400

600

800

I000

R(~,)

Fig. 6. LOgloM(e q. (17)) as a function of R, IOA<~R <<.1000 A, for x = 0.10 cm, x = 0.20 cm, and t = 20 sec. electron microscopy. This function is shown in fig. 6 for x = 0.10 cm and x = 0.20 cm. It is seen that for small R, there is a significant difference between the two curves. It should also be noted that despite the substantial degree of crystallinity indicated in fig. 2, so much of this crystallinity consists of tiny crystals having R ~< 10 A that relatively few crystals will be observed in transmission electron microscopy. This example illustrates the detailed character of the information which can be obtained by use of the crystal distribution function, ~. The specific forms of the results shown in figs. 1 - 6 depend strongly upon the assumed quenching treatment (and hence thermal history) as well as upon the assumption of homogeneous nucleation. Observations of crystal distributions in specimens given a particular quenching treatment can in turn be used with the present analysis to provide information about the distributions of nucleating heterogeneities in a specimen; and experiments directed to this end will soon be carried out.

Acknowledgements Financial support for the present work was provided by the National Aeronautics and Space Administration and by Owens-Illinois Inc., who provided one of the authors (G.S.) with the Owens-Illinois Fellowship in Materials Science. This support is gratefully acknowledged.

Appendix A N o n i s o t h e r m a l diffusion controlled g r o w t h

In general, isothermal diffusion controlled growth is described by the equations

R.W. Hopper et al., Crystallization statistics, thermal history and glass formation [D(C)C~]~ + o(t)C~ = Ct ( ~ > O, t > 0),

57

(A.1)

o) = c o ,

(A.2)

c( o, t) = c o ,

(A.3)

C(0, t) = CLe

(A.4)

D [C(0, t)] C~(O, t) + o(t) (CLe - CSe) = 0.

(A.5)

Here C is the concentration per unit volume, CSe is the equilibrium solid concentration, CLe is the equilibrium liquid concentration, and ~ is a spatial coordinate moving with the interface, t

=f v(t')dt'.

(A.6)

0 Eq. (A.1) is just the diffusion equation of solute in the liquid, eq. (A.2) is the initial condition, eq. (A.3) is the boundary condition, and eq. (A.5) is a conservation of solute equation. Eq. (A.4) represents the basic assumption of diffusion-controlled growth, that the solid and liquid are in equilibrium at the interface. The imposition of four subsidiary conditions on eq. (A.1) uniquely determines the unknown growth velocity, o(t). It is easily shown that o(t) = A t -1/2. In nonisothermal diffusion-controlled growth, the temperature is a function of time but not of position IT = T(t)], and the limiting growth rate at all times is determined by diffusion of solute and not by interface attachment kinetics. The same set of equations may be applied, with D(C) replaced by D [C, T(t)] in eq. (A. 1), and CLe and CSe by CLe [T(t)] and Cse [T(t)] in eqs. (A.4) and (A.5). Since T(t) is a known function of t, the new set of equations are eqs. (A.2), (A.3), (A.6), and [D(C, t)C~]~ + v(t)C~ = Ct ,

(A.7)

C(0, t) = CLe(t),

(A.8)

D [C(0, t), t] C~ (0, t) + o(t) [Cte(t ) - Cse(t)] = 0.

(A.9)

This set of equations must now be solved to obtain o(t). This is often quite difficult. The principles of sect. 2 can be applied directly if v(t) is known. Note, however, that v(t) will depend upon the time when growth begins, since T(t) is independent of the time of nucleation. Thus, since T(t) depends on r, we write V = V(r, t O; t). With this change, the equations of sect. 2 all hold, except that u IT(r, t)] is replaced by the explicit function V(r, to; t). Thus, one obtains

58

R.W. Hopper et aL, Crystallization statistics, thermal history and glass formation

I v {Tit, to(r, t, R)] } V(r, to; to)

~(r, t, R ) =

(A.IO)

where t

R(r, t, to) = f

V(r; to; t') dt' (R > 0)

(A.11)

to

determines to(R ) .

Appendix B Transient nucleation

In this appendix, the results of sect. 2 are modified to allow for delay times (transient times) in nucleation. These times, which represent the period during which the sub-critical sized embryos build up toward their steady-state distribution (and the nucleation frequency correspondingly l~uilds up toward its steady-state value), can be important in some glass-forming systems. In assessing their possible importance, the delay times in nucleation must, however, be compared with the overall transformation (crystallization) time; only when they are significant relative to the overall time should the nucleation delay times be regarded as significant for the overall crystallization process. As in appendix A, the difficulty in including the delay times lies not with the analysis of the crystallite statistics but in the determination of the nucleation rate as a function of time when the temperature varies with time. Standard nucleation theory (e.g. [13]) considers the formation of crystalline nuclei of a critical size in a liquid. When T-is below the equilibrium melting point TE, a large cluster of crysthlline material is stable and will grow in time. Very small crystalline clusters are unstable, however, because of their relatively large surface free energy. Nucleation theory considers the rate at which stable clusters, or 'nuclei', are formed in a series of bimolecular reactions. K+ A m + A ° m, A m + l . (B.1) Here A m represents a cluster of m molecules. Once this sequence of reactions has reached steady state, the steady-state nucleation frequency is IvSS(T). Following an initial quench from T 1 > TE to T < T E, however, there is an initial transient time during which the steady-state concentrations of the sub-critical clusters are established. This problem has been studie.d by a number of authors, including Zeldovich [ 14], Turnbull [ 13] and Russell [ 15]. One must obtain the time-dependent solution of the system of equations

R. W. Hopper et al., Crystallization statistics, thermal history and glass formation

dXm dt - K + - I X m - 1

+ K m + l X m + l - (K+ +Km)Xm '

59

(B.2)

where X m is the concentration o f A m . While no general solution is available, one can as an approximation, assume [ 13-15] a single characteristic relaxation time r such that Iv(t) = e -t/r I ss V

(B.3) '

where the delay time r for isothermal nucleation is given to perhaps order of magnitude by 7" ~ (n *)2/Ns~'.

(B.4)

Here n* is the number of molecules in the critical nucleus, N s is the number of molecules on the surface of the critical nucleus, and ~, is the frequency of molecular transport across the nucleus-liquid interface. The expression of eq. (B.3) represents a rough approximation to the overall time dependence of the nucleation frequency. It is an excellent approximation in the early stages of isothermal nucleation (Iv(t) "~ IvSS), however, and the r of eq. (B.4) should provide a useful estimate of the delay time before the steady-state nucleation rate is established. The nonisothermal problem, in which the rate constants and the boundary conditions (i.e., the critical nucleus size) of eq. (B.2) vary in time has not been studied. With the aid of modern digital computers, one could attack the problem with success and obtain the Iv(t ) desired for the present applications. The equations of sect. 2 can be easily modified if it is assumed that I v = Iv(r, t). The basic arguments still hold, and one obtains Iv[r, to(r, t,R)] ~(r, t, R ) - u{T[r, t o (r, t, R ) ] )

(B.5)

where t o is given implicitly by eq. (2).

Appendix C E x c l u d e d volume effects

In the analysis of sect. 2, excluded volume effects were ignored; that is, the fact that some of the volume element do has previously transformed and is therefore unavailable for nucleation events was omitted from eq. (3). It was also assumed that the crystallites do not overlap. In this appendix, these restrictions are removed. In a classic analysis, Avrami [ 2 - 4 ] treated the problem of the volume fraction transformed as a function of time when overlap and excluded volume effects are

60

R. W. Hopper et aL, Crystallization statistics, thermal history and glass formation

\

Fig. C. la. Morphology in which R is not a useful parameter and in which qJ is not well-defined. taken into account. In the present n o t a t i o n , it is found that the volume fraction transformed at (r, t) is Fv(r, t) = 1 - exp

_ f

54 rrR (r, t, t0)3 1 v [T(r, to) ] dt 0

.

(C.1)

w--

C. lb. Morphology having high volume fraction crystaUinity, crystaUite overlay and important excluded volume effects but in which R is a useful parameter. Fig.

R. W. Hopperet al., Crystallizationstatistics, thermalhistory and glassformation

61

This result can be used to correct eq. (3) for excluded volume effects in nucleation. The volume of uncrystallized material is [1 - F v ( r , to) ] do, and the number of crystallites nucleated between t o and t o + dt 0 is dn = Iv(t0) exp f -

fOt 47rR(r, to,

t') 3 1v [T(r, t')] dt' } do dt 0 .

(C.2)

The succeeding steps remain valid, and one obtains

~(r,t, Ro)-

Iv[T(r' t°)] u[T(r,

to) ]

{ exp

- f

t°

0

~nR(r, to, t')3Iv[T(r,t')]dt'

)

.

(C.3)

Here R 0 and t o are related by R 0 = R(r, t, to) in eq. (2). One could, with the aid of eq. (5), change variables from t' to R in the integral of eq. (C.3), but the result is not particularly useful. Eq. (C.1) includes the overlap of growing crystals-i.e., the fact that impingement stops growth over a common interface. Eq. (C.1) therefore replaces both eqs. (10a) and (10b) and is generally valid. When impingement effects are sufficiently severe that R as defined by eq. (2) is no longer meaningful, the analysis breaks down. Such a case is shown in fig. C. la. In other cases R remains a useful dimension despite impingement, as in fig. CAb. In such cases, both ff and D N are meaningful functions. Eq. (C.3) gives qJ; and with this expression, eq. (9a) remains valid. Eq. (9b) must, however, be replaced by

DN(r,t): f 'lv[T(r, to)]exp 0

{? -

~nR(r, to, t')31v[T(r,t')]dt'

} dt o.

(C.4)

0

Eq. (C.4) is valid even when impingement is substantial.

References [1] [2] [3] [4] [5] [6] [7]

W.A. Johnson and R.F. Mehl, Trans. AIME 135 (1939) 416. M. Avrami, J. Chem. Phys. 7 (1939) 1103. M. Avrami, J. Chem. Phys. 8 (1940) 212. M. Avrami, J. Chem. Phys. 9 (1941) 177. D.R. Uhlmann, J. Non-Crystalline Solids 7 (1972) 337. R.W. Hopper and D.R. Uhlmann, J. Crystal Growth 19 (1973) 177. R.W. Hopper and D.R. Uhlmann, Crystallization Statistics and Thermal History of Bodies Cooling under Radiation Conditions, to be published. [8] M. Cukierman, P.M. Tutts and D.R. Uhlmann, in: Proc. of the Third Lunar Science Conf. (Suppl. 3. Geochim. et Cosmochim. Acta), Vol. 3 (MIT Press, Cambridge, 1972) p. 2619. [9] G. Scherer, R.W. Hopper and D.R. Uhlmann, in: Proc. of the Third Lunar Science Conf. (Suppl. 3, Geochim. et Cosmochim. Acta), Vol. 3 (MIT Press, Cambridge, 1972) p. 2627. [10] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford University Press, Oxford, 1959) p. 88.

62

R.W. Hopper et aL, Crystallization statistics, thermal history and glass formation

[11] D.R. Uhlmann, in: Materials Science Research Vol. 4 (Plenum Press, New York, 1969) p. 172. [12] W.B. Hillig, in: Reactivity of Solids (Wiley, New York, 1969) p. 639. [13] D. Turnbull, in: Solid State Physics, Vol. 3 (Academic Press, New York, 1956) p. 225. [14] J.B. Zeldovich, Acta Physiocochem. USSR 18 (1948) 774. [15] K.C. Russell, Acta Met. 16 (1968) 761.