Overview of crystallization in fluoride glasses

]OURNA

Journal of Non-Crystalline Solids 140 (1992) 1 - 9 North-Holland

L OF

NON-CRYSTALLINE SOLIDS

Overview of crystallization in fluoride glasses Marcel Poulain Centre d'Etude des Mat~riaux Auancds, Laboratoire de Chimie Min~rale, Uniuersit~ de Rennes-Beaulieu, Avenue du Gdndral Leclerc, 35042 Rennes Cddex, France

Most fluoride glasses are likely to devitrify when heated for some time between the glass transition and melting temperatures. Their crystallization has been widely studied in order to m e a s u r e kinetic parameters and to ease their development. Several experimental T T T curves have been reported, providing a global picture of the thermal instability range. Critical cooling rates have been deduced from systematic measurements. The J o h n s o n - M e h l - A v r a m i approach provides theoretical relations which provide the basis for isothermal and non-isothermal m e t h o d s using differential scanning calorimetry. T h e value of the Avrami exponent, n, and activation energy, E, for various fluoride glasses have been deduced from experimental measurements. They suggest that crystals grow tridimensionally from a constant n u m b e r of nuclei. However, this m e c h a n i s m appears different and probably more complex in fluorochloride and less stable fluoride glasses. Nucleation around Tg appears very limited in the general case. Activation energy as measured by DSC is correlated with the difference between the crystallization temperature and Tg. T h e physical meaning of kinetic parameters is discussed. A m o n g other questions, the role of anionic oxygen in nucleation and crystallization processes needs to be known more precisely.

1. Introduction Crystallization processes provide both the reference and the nightmare of glass scientists. Among other characteristics, glass appears first as a non-crystalline solid. When trying to synthesize a new glass, the lack of crystals - easily visible under polarized light - is the first indication of the vitreous state. Much more rapid than X-ray diffraction, visual observation is also far more sensitive. On the other hand, the accurate structural data collected from crystallography make the basis for most descriptions of glass structure, as short range arrangements are very similar for both crystalline and vitreous materials. The crystallization rate of the melt during cooling defines an accurate basis for glass-forming ability, and the general approach which defines glass as a material which did not have time enough for crystallization applies universally. However, some haunting questions are still waiting for satisfactory answers. For example,

what is the relation between glass-forming ability and chemical factors (anionic radii, bonding, covalency, electronegativity), and thermodynamic and structural features of the elements involved in a glass composition. More basically, how does one prevent crystal formation? Time and temperature govern the transition from liquid to crystal. Unfortunately, our understanding is still incomplete leaving room for empiricism and intuition.

2. The place of crystallization in the development of fluoride glasses Unexpected in their occurrence, fluoride glasses are being developed for potential applications only because they show a minimum stability allowing optical elements to be tested on the experimental scale. Actual requirements for practical optical components imply that slow cooling rates may be used for glass preparation. The situation is different for bulk components and optical fibres.

0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

2

M. Poulain / Crystallization in fluoride glasses

2.1. Bulk components The manufacture of high quality lenses, prisms and windows in fluoride glasses faces numerous problems. Among them, one may quote inhomogeneity resulting in refractive index variations, and random bubbles and crystals which often grow along isothermal surfaces at some stage of the solidification. It is often difficult to ensure good reproducibility of the selected physical properties: n D or thermal expansion. In most cases, slow cooling rates and careful annealing are required. Unfortunately, these conditions also favour nucleation and crystal growth. To a large extent, mechanical properties are influenced by uncontrolled crystals, especially near surfaces. Finally, all these problems become more critical as sample size increases. In summary, fabrication of large fluoride glass optics requires very stable glass compositions, and rigorous processing which achieves a convenient balance between slow cooling and nucleation control.

2.2. Optical fibres Fluoride glass optical fibres are currently drawn from preforms, which means that the glass crosses twice the thermal range of instability: first on cooling for preform manufacturing and then on reheating for fibre drawing. For single mode preforms, additional operations are carried out in the softening range [1] which increases the probability of nucleation. In the early stage of research on fluoride glass fibres, drawing from melt, using the crucible method appeared, a priori, more favourable. Results did not confirm this expectation [2,3] since crystallization is more important with this process, viscosity control is very difficult, and a significant part of the melt must be kept in the instable thermal range. Finally, crucible methods require a glass still more stable than the preform method in order to prepare low loss optical fibres. This is unfortunate because fibre drawing from the melt allows the continuous manufacturing of long lengths, resulting in lower costs. The need for long lengths which will derive from the achievement of low optical losses requires that preform size is substantially increased,

allowing tens of km to be drawn from one single preform. This could be achieved only if crystallization processes can be controlled. Practical fibres have been found to exhibit random defects, either internal or external, which severely reduce tensile strength and increase dramatically optical losses. The numerous studies devoted to low loss optical fibers emphasize the critical role of defects in optical losses. Depending on defect size, scattering losses decrease with wavelength as A 4 or ~ 2 or remain wavelength independent. This later behaviour is found to be the main factor limiting the transmission of fluoride glass optical fibres. It appears still more critical for single mode optical fibres for which the contribution of interface defects to overall losses may be predominant, insofar as their small core must be free of crystals inducing MIE or wavelength independent scattering. Crystallization at the interfaces - e.g. core/cladding or cladding/atmosphere - usually occurs before bulk crystallization. Fibres in which index matching between core and cladding is made by PbF 2 doping often exhibit crystals at the interface, probably because intermediate glass compositions are intrinsically less stable. The potential of fluoride glasses for ultratransparent fibres will ultimately depend on the possibility of controlling scattering losses. Whether the defects inducing these losses are intrinsic or extrinsic in nature is of prime importance. While it has been reported that observed losses are consistent with expected nucleation levels [4], other studies emphasize the critical role of anionic oxygen. First, it has been observed that scattering losses increase almost linearly with oxygen content [5]. Second, critical cooling rate is also strongly correlated with residual oxygen [6]. This rather suggests that heterogeneous nucleation is often predominant and there are several observations in favour of this assessment. Depending on processing, glasses of the same nominal composition may show different crystallization behaviour. Also, using rigorous conditions of synthesis and very clean and dry enclosures, optical fibres may be drawn from glasses which exhibit a DSC exotherm at the drawing temperature. Finally, the fact that low optical losses have been mea-

a/L Poulain / Crystallizationin fluoride glasses sured on short length fibres [7-9] obviously implies that the defects observed otherwise are extrinsic. It is also consistent with recent calculations of scattering losses in fluoride glasses [32]. Nevertheless, the search for glass compositions more stable against devitrification should remain a major objective•

3

been reported to express this correlation via a first order approximation:

1 -ln(-ln(a n

-x))

= In

1 --ln(-ln(1

-x))

=in-

n

koR E

Tff E + In-- - -a

R~'

koE

E

dR

RTp

(3)

E -21n--

RL (4)

3.

Basic

relations

Nucleation and crystal growth can be expressed by a n u m b e r of relations deriving from theoretical approaches. The relation defined by Avrami [10] which forms the basis of numerous studies is written as x = 1 - exp(-kt)",

(1)

where x is crystalline fraction, t is time, k is a factor which accounts for both nucleation and crystal growth rates and n is an exponent which depends on crystallization mechanism• This relation is strictly only valid for isothermal transformations in which one single phase is growing. In a limited t e m p e r a t u r e range, the dependence of k on t e m p e r a t u r e may be described as Arrhenian: k = k o exp(-E/RT),

(2)

where E is the activation energy for crystal growth and nucleation. This relation, which is widely used, has been introduced via two different theoretical analyses [12,13]. Thus, the Avrami exponent, n, and activation energy, E, appear as kinetic factors governing the devitrification process. The frequency factor, k0, has attracted less attention, probably because its value is often much less accurate. While DSC isothermal m e a s u r e m e n t s provide access to n and E, other relations have been derived in order to deduce n and E from D T A or DSC scans. Assuming that the crystalline fraction, i.e. x, has always the same value when an exothermic p e a k reaches its maximum value, the shift of the corresponding temperature, Tp, is correlated with heating rate, or. Using the same notation as above, the following relations have

According to relation (3), a plot of i n ( T Z / a ) versus Tp-1 should yield a straight line with a slope of E / R [11]. Alternatively, relation (4) defines a linear variation of In a versus Tp 1 with a slope equal to - E / R [14], as l n ( E / R T p ) may be neglected• In practice, any plot of ln(a/Tpm) versus Tp 1 appears linear for m values ranging from 0 to 2 [17], and the limited experimental T range makes it difficult to assess any significant deviation from linearity. Indeed, a set of calculations using the different formulae with the same data lead to similar values of the activation energy [18]. Relation (4) provides also a method for the determination of the Avrami exponent n. For a constant value of temperature, T, the depen: dence of l n ( - l n ( 1 - x ) ) on a is linear and the slope is equal to - n . Calculation is easily achieved by measuring crystallization fraction, x, for different heating rates at a given temperature. A similar equation has been proposed by other authors using a starting point other than Avrami's relation [15,16]. Time-temperature-transformation (TTT) curves may be drawn from experimental measurements. These are described by the relation

x ~ (~r/3)lvu3t 4,

(5)

where t is time, l v is the nucleation rate per unit volume and u is the crystal growth rate [19]• The critical cooling rate, CCR, may be roughly estimated as CCR

= ( T L --

Ty)/t y

where T L is the liquidus temperature, and T N and t N are the t e m p e r a t u r e and time at the nose of the T T F curve. Recent work provides a more accurate relation for C C R calculation [20] which

4

M. Poulain / Crystallization in fluoride glasses

requires t h a t the evolution of I v and u versus t e m p e r a t u r e is known. It is also possible to estimate the critical cooling rate, Rc, from a set of D T A or DSC scans during cooling using the equation [21,22] In a = in R e - b / A T ~ , where b is a constant, and AT~ is the difference between liquidus t e m p e r a t u r e and crystallization t e m p e r a t u r e for melt cooling. The critical cooling rate is obtained by an extrapolation of the linear plot In a vs 1 ~ A T 2 towards 1 ~ A T 2 = O.

4. Results 4.1. T T T curves

Time-temperature-transformation curves have been drawn from experimental results using

440

DSC. The method consists of setting the DSC cell at a constant t e m p e r a t u r e while glass is quickly melted in a DSC pan. Then molten glass is transferred into the DSC cell and the thermal signal is recorded. After thermal equilibrium is reached, heat flow is proportional to crystallization rate. The maximum of the exotherm which corresponds to a constant crystallization fraction, e.g. 45%, defines both a t e m p e r a t u r e and a time, allowing a T T T curve to be drawn after a set of measurements at different temperatures. Examples of such curves are reported in fig. 1 [13,23]. Other T T T curves have been drawn for fluorochloride [24] and fluoroindate [25] glasses. Some care must be taken in the use of such curves for the following reasons: first the transformation rate, close to 50%, corresponds to a material which does not have the expected characteristics of a glass. Times for a reduced crystalline fraction, e.g. 10 - 4 t o 10 - 6 , should be

440

a

420

420

~a

400

400

[... ,,¢ "< k~

380

380

360

"< 360 340

340 [.

[-

320 1.2

214

216

218

3.0

3.2

320

3.4

2.2

Log t ( seconds )

520 e

440

~ ~

~

420

~

400

~ [.<

380

420

~.

340

400

~

320

480

460 440

2.2

2.4t

2'.6

2'.8

2.6

2.8

3.0

3.2

3.4

i 3.0

3.2i

i 3.4

Log t ( seconds )

d

500 ",~ m m [.. ,< ee

2.4

~.0

Log t ( seconds )

3.'2

~.4

(..

360

i 2.2

i 2.4

2'.6

i 2.8

Log t ( seconds )

Fig. 1. T i m e - t e m p e r a t u r e - t r a n s f o r m a t i o n curves for s t a n d a r d f l u o r o z i r c o n a t e glasses Z B L A L (a), Z B L A N (b), Z B L A (c) and Z B N A (d). O p e n circles r e f e r to the s m a l l e r crystallization fraction c o r r e s p o n d i n g to o n s e t of e x o t h e r m . T h e lines are d r a w n as a g u i d e for the eye.

34, Poulain / Crystallization in fluoride glasses Table 1 Critical cooling rates Rc reported for various fluoride glasses Glass:

ZBG

ZBGA

ZBLA

HBLA

ZBLAN

ZBLYAL

YABC

R c (K/rain):

350

40-70

55

60-180

3

25

Ref.:

[22]

[6,22]

[61

[6]

[6]

[22]

much smaller. On the other hand, melting and handling a small piece of glass at room atmosphere induces hydrolysis and contamination, resulting in a heterogeneous nucleation much higher than in current glass samples. Such heterogeneous nucleation may be responsible for the observed crystallization tendency in the upper part of the T T T curve. In practice, these two factors counteract each other, so that the times for crystallization and the deduced critical cooling rates are consistent with experimental observations. More reliable data would require that all operations are carried out in a very dry and clean enclosure, and that the influence of surface nucleation for small samples is evaluated.

4.2. Critical cooling rate measurements Using a method described by Barandarian and Colmenero [21], the critical cooling rate of a glass may be evaluated from D T A measurements. Table 1 gives some values of critical cooling rates obtained in this way [22,26]. Typical values are in the range of tens of K / m i n , but may depend on heterogeneous nucleation factors, especially residual oxygen.

BIZYbT

BIZYTG

200-700

120

20

[6]

[26]

[26]

4.3. Kinetic parameters of crystallization Numerous studies have concentrated on the determination of Avrami exponent, n, and activation energy, E. The physical meaning of these parameters is generally assessed as follows. (1) n depends directly on the mechanism of crystal growth (mono-, bi- or tridimensional) and the factor controlling crystallization: diffusion or interface. Table 2 summarizes various possibilities [27]. (2) The activation energy for crystallization is the same as for shear viscosity at some temperature (3) Lower activation energies correspond to more stable glasses. Most studies implemented on fluorozirconate glasses lead to n values close to 3 [16-18,28] which are consistent with tridimensional growth from a constant number of nuclei. Activation energies lie in the range 40-100 kcal/mol. Similar measurements on a fluoroindatc glass give the following results: n = 3.2 and E = 60 kcal/mol. Typical plots are reported in fig. 2. Recent work centered upon fluorochloride glasses suggest that real processes may be more complex. While isothermal and p s e u d o - i s o t h e r -

Table 2 Possible values of Avrami exponent, n [27] Conditions Interface controlled growth and increasing nucleation rate Constant nucleation rate Decreasing nucleation rate Zero nucleation rate (or saturation of point sites) Edge nucleation after saturation

n

Conditions

n

>4 4 3-4

Diffusion controlled growth and increasing nucleation rate Constant nucleation rate Decreasing nucleation rate Zero nucleation rate

>2.5 2.5 1.5-2.5 1-1.5

Needles and plates

6

M. Poulain / Crystallization in fluoride glasses

mal methods provide similar values for n in fluorozirconate and fluoroindate glasses, significant variation may be observed in other cases. For example in the 50ZrF4-40BaF2-5ThF4-3LaF 312MC1 glass (M = Na, K, Rb), the value of n obtained by the pseudo isothermal method could vary from 2.7 to 5.5 depending on temperature [24,29]. This leads to the conclusion that the crystallization mechanism may be different when temperature increases or that several mechanisms occur simultaneously. In this case, the Avrami relation may not apply, because it assumes that there is only one liquid/crystal transformation, or at least transformation of the same type. The complexity of the phenomenon may be exemplified by the dependence of the isothermal crystallization peaks on temperature in a fluorochloride glass (fig. 3). The evolution of activation energy versus composition may provide useful information on the optimum glass compositions. Figure 4 reports an example of such a plot for a fluoroindate glass, suggesting a maximum stability at 38% InF 3 and a less stable glass at 35% InF 3 [29].

© .2

,<:

1

I

I

I

I

I

I

10

20

30

40

50

60

70

T IM E

80

( minutes )

Fig. 3. Evolution of the exothermic peaks versus crystallization temperature. Glass composition is 50ZrF4-30BaF 2 5ThF4-3LaF3-12RbC1.

4.4. Nucleation However, the assumption that crystal growth occurs from a constant number of nuclei implies either that all nuclei are generated during cool-

Homogeneous nucleation is expected to be predominant in most unstable fluoride glasses. 0

-2.6 ----

.

r--

°

-2.8 " -1

,-~

&

--

=

-3.0 -

-3.2 (a) -3

•

1.6

(b) •

,

1.8

•

.

,

.

2.0

•

,

2.2

In ( O~)

.

•

,

2.4

.

•

-3.4

2.6

1.48

1.49

1.50

1.51

1.52

(10~ Tp) K "1

Fig. 2. (a) Typical plots of log ~ versus 1000/ATe, where o: is beating rate and ATc is the difference between liquidus temperature and crystallization temperature at cooling. (b) Determination of Abrami exponent, n, and activation energy for a fluoroindate glass (IZBS) using pseudo-isothermal and non isothermal methods.

M. Poulain / Crystallization in fluoride glasses 80

70

e-

60

3¸

<

50

2;

;o

i

3'0

i

40

50

60

7

at a constant temperature; second, a DSC scan at 10 K / m i n . If extra nuclei are formed, then more numerous crystals will grow; therefore the crystalline fraction is larger and the exothermic peak is shifted toward lower temperatures. In most cases, no significant shift is observed before the annealing t e m p e r a t u r e becomes close to the crystallization temperature. Typical results are shown in fig. 5~ This suggests that low t e m p e r a t u r e nucleation is a rather limited phenomenon, at least when time does not exceed a few hours. Previous attempts of sub-Tg annealing for several months did not show significant change in the DSC scans of fluorozirconate glasses [31].

% InF3 Fig. 4. Evolution of activation energy as a function of InF 3 content in the ( 6 0 - x ) Z n F 2 2 0 B a F 2 - 2 0 S r F 2 - x l n F 3 glass. The line is drawn as a guide for the eye.

ing, some time below the liquidus temperature, or that nucleation is negligible when crystal growth is significant. A set of experiments has been implemented in order to determine if significant nucleation could occur near Tg. Measurements include two steps: first, annealing for a constant time (e.g. 20 min) 400

[]

[]

[]

390

o

380

370

360 280

3~0

3~0 T.

Iso

3;0

3~0

(°C)

Fig. 5. Evolution of the DSC exotherm peak temperature, Tw as a function of annealing temperature, Tiso, for 20 rain. Glass composition is 30InF3-30ZnF2-20SrF2-15BaF2_5CdF2. The line is drawn as a guide for the eye.

5. Discussion

As pointed out in the introduction, crystallization studies encompass different fields of interest. For basic science researchers, they provide a set of information which aids the understanding of glass formation. In this respect, fluoride glasses appear intermediate between major oxide glasses and metallic glasses for which timescales for devitrification are either much longer or much shorter. Some areas remain less explored, such as precise determination of liquidus temperatures, surface nucleation and identification of crystalline phases. Also, the physical meaning of activation energy is still far from obvious. Because it is assumed to be the same as for shear viscosity, one may assert that there is a mechanism which is common to viscous flow and crystallization. It would seem logical to correlate the activation energy and the energy of chemical bonds which are broken - and recombined - in the molten state. Unfortunately, higher activation energies are observed near Tg, i.e., when weaker chemical bonds are affected. A dynamic structural model appealing to fractal concepts could be helpful, the more so as it could be tested by computer simulation. Most crystallization studies of fluoride glasses have been made for practical reasons in relation to their potential applications. In the search for new vitreous compositions, it is useful to assess quickly and simply stability against devitrification,

8

M. Poulain / Crystallization in fluoride glasses

in order to define the composition adjustments more likely to lead to low nucleation rate glasses. D T A or DSC scans require small samples and little time. For this reason, various stability criteria using characteristic temperatures have been used. A m o n g them, the separation between glass transition and crystallization temperature, Tc-Tg, is considered a sensitive p a r a m e t e r available from a single D T A measurement. However, it cannot define a reliable stability scale. For this purpose, critical cooling rate is certainly more appropriate, although it requires a set of DSC measurements, as for T T T curves. As reported in the previous section, lower activation energies, E, usually correspond to more stable glasses, and this correlation may be used for assessing the best composition range in a complex glass formulation. It was first assumed that these low values of E, which are also related to those of viscous flow, indicate a viscosity profile more favourable to glass formation. When plotting log r/ versus 1/T or, in a normalized way, versus Tg/T, a strong deviation from linearity is observed in fluoride glasses, as shown in fig. 6. Activation energy corresponds to the derivative of this curve, expressing the value of the slope at any point of the plot. Consequently, it decreases significantly as t e m p e r a t u r e increases from the glass transition to the liquidus tempera-

Tm ~

~ Tg

10

0.5

1

Tg / T Fig. 6. Schematic viscosity profile log ~7= f ( T / T g ) for common fluoride glasses. The dashed line expresses ideal Arrhenian behaviour. Derivative curve gives activation energy.

ture. If we r e m e m b e r that, in the non-isothermal method, activation energy is calculated from the shift of the crystallization peak, it becomes obvious that a low value of activation energy reflects a large separation between crystallization temperature and Tg. In other words, E is strongly correlated with the difference Tc-Tg. Further work is needed in order to assess to what extent more information about glass stability can be deduced from its absolute value. From the value of the Avrami exponent, n, one may deduce the possible mechanisms of crystal growth. There are still some uncertainties about the accuracy of the results as it may be sensitive to the profile of the recorded exotherm which may be dependent on the DSC set-up. Moreover, it is not certain that the Avrami relation describes completely and accurately the entire process occurring in some multicomponent glasses. It is surprising that the n value is not found to change when fluoride glass samples are powdered [16], increasing significantly the specific surface and shifting the crystallization peak; one would expect simultaneous surface and bulk crystallization resulting in a lower mean n value. Nevertheless, it is observed that, in stable fluoride glasses, n is ~ 3 while attempts for drawing fibres from glasses for which n > 3 did not succeed. There are still some interesting questions in relation to crystallization. First, what is the influence of crucible on nucleation? While there is no evidence at the m o m e n t for any nucleating action of platinum, it could be interesting to know more about m e t a l - m e l t interaction. The exchanges between melt and atmosphere, either dry, neutral or reactive, make also an important field of investigation as they often result in heterogeneous nucleation. Finally, the exact role of anionic oxygen in devitrification is not known. The formation of oxide crystals has been reported. Are these crystals sufficient to induce wavelength independent scattering? Or do they act also as nucleating agents? Is oxygen concentration roughly constant in bulk glass? The answers to these questions will bring new questions and finally will open the way towards ultra low loss fluoride glass optical fibres.

M. Poulain / Crystallization in fluoride glasses

6. Conclusion T h e m a j o r r o l e o f n u c l e a t i o n a n d crystallizat i o n p r o c e s s e s in t h e d e v e l o p m e n t o f f l u o r i d e glass o p t i c a l c o m p o n e n t s is c l e a r l y p e r c e i v e d . F o r f l u o r i d e glasses, t h e c r i t i c a l c o o l i n g r a t e r e p r e sents one major parameter defining the intrinsic l i m i t s o f c u r r e n t m a n u f a c t u r i n g p r o c e s s e s . It m a y be estimated from experimental TTT curves, or p e r h a p s m o r e c o n v e n i e n t l y by t h e m e t h o d d e s c r i b e d by B a r a n d a r i a n a n d C o l m e n e r o . I n p r a c tice, t h e c o n t r o l o f h e t e r o g e n e o u s n u c l e a t i o n app e a r s still m o r e c r i t i c a l t h a n i n t r i n s i c glass stability [32]. Various non-isothermal methods based on D T A o r D S C s c a n s give a c c e s s to t h e a c t i v a t i o n e n e r g y a n d e x p o n e n t n, w h i c h a r e c o n t a i n e d in the Avrami equation. To some extent, these par a m e t e r s r e f l e c t glass stability, b u t t h e i r p h y s i c a l m e a n i n g is n o t always o b v i o u s . S e v e r a l q u e s t i o n s in r e l a t i o n to c r y s t a l l i z a t i o n a r e still p e n d i n g , especially the chemical transformations induced by a n i o n i c o x y g e n .

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9

[8] G. Lu and I. Aggarwal, Mater. Sci. Forum 19-20 (1987) 375. [9] S.F. Carter, M.W. Moore, D.S. Zebesta, J.R. Williams, D. Ranson and P.W. France, Elect. Lett. 26 (1990) 2015. [10] M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. [11] H.S. Chen, J. Non-Cryst. Solids 27 (1978) 257. [12] H.E. Kissinger, J. Res. Natl. Bur. Stand. 57 (1956) 217. [13] M.C. Weinberg, J. Non-Cryst. Solids 127 (1991) 151. [14] T. Ozawa, Bull. Chem. Soc. Jpn. 38 (1965) 1881. [15] K. Matusita and S. Sakka, Phys. Chem. Glasses 20 (1979) 81. [16] D.R. MacFarlane, M. Matecki and M. Poulain, J. NonCryst. Solids 64 (1984) 351. [17] D.R. MacFarlane, in: Nucleation and Crystallization in Fluoride Glasses, ed. A.E.Comyns (Wiley, New York, 1989) p. 49. [18] M.A. Esnault-Grosdemouge, M. Matecki and M. Poulain, Mater. Sci. Forum 5 (1985) 241. [19] D.R. Uhlmann, J. Am. Ceram. Soc. 66 (1983) 95. [20] M.C. Weinberg, B.J. Zelinski and D.R. Uhlmann, J. Non-Cryst. Solids 123 (1990) 90. [21] J.M. Baradarian and J. Colmenero, J. Non-Cryst. Solids 46 (1981) 277. [22] T. Kanamori and S.J. Takahashi, J. Appl. Phys. 24 (1985) 1758. [23] L.E. Busse, G. Lu, D.C. Tran, and G.H. Sigel, Mater. Sci. Forum 5 (1985) 219. [24] A. Elyamani, Th~se de doctorat, University of Rennes (t989). [25] Y. Messaddeq, Th~se de doctorat, University of Rennes (1990). [26] I. Chiaruttini, Th~se de doctorat, University of Rennes (1990). [27] J.W. Christian, The Theory of Transformation in Metals and Alloys, 2nd Ed (Pergamon, Oxford, 1975). [28] N.P. Bansal, R.H. Doremus, C.T. Moynihan and A.J. Bruce, Mater. Sci. Forum 5 (1985) 211. [29] A. Elyamani, M. Matecki and M. Poulain, Mater. Sci. Forum 67-68 (1991) 211. [30] Y. Messaddeq, M. Matecki and M. Poulain, Mater. Sci. Forum 67-68 (1991) 251. [31] M.A. Esnault-Grosdemouge, Th~se de doctorat, University of Rennes (1984). [32] G. Lu, P. Hart and I. Aggarwal, Phys. Chem. Glasses 31 (1990) 205.