J. Chem. Soc., Faraday Trans. I, 1981, 77, 2297-2307

Kinetics of Polymorphic Transitions in Tetrahedral Structures

Part 2.-Temperature Dependence of the Transition /? y Li,ZnSiO,

BY M A R I A E. VILLAFUERTE-CASTREJON~ Colegio de Ciencias y Humanidades, Plantel Oriente, Unidad Academica Ejerci to

de Oriente, Mexico 9DF, Mexico A N D ANTHONY R. WEST*

University of Aberdeen, Department of Chemistry, Meston Walk, Old Aberdeen, Aberdeen AB9 2UE

Received 22nd November, 1979

The kinetics of the transition y B Li,ZnSiO, have been studied in the range 480-855 OC and in the reverse direction over the range 906-940OC. X-ray powder diffraction was used to determine the degree of transformation in powdered samples annealed isothermally for various times. At most temperatures, the data could be analysed using the Johnson-Mehl-Avrami equation and a rate constant value extracted. At two temperatures, 480 and 855 OC, the data appear to fit an autocatalytic rate law. The effect of particle size on transformation rate was studied at one temperature, 572 O C ; the rate increased with decreasing particle size. An Arrhenius plot of rate constant data was constructed; below cu. 780 O C ,

a straight line was obtained for the y -, /? transformation with an activation energy of 147 f 20 kJ mol-'. As the equilibrium transition temperature, 880 f 20 OC, was approached the transition rates in both directions decreased increasingly rapidly. In the range ca. 860-900 O C no detectable transformation in either direction occurred for heating times of up to two weeks. The temperature dependence of the y + jl transition rate has been modelled satisfactorily by regarding the transformation as a modified, microscopically reversible transformation at all temperatures, subject to the constraint that macroscopically the phase rule must be obeyed, The rate data have also been presented on a time-temperature-transformation diagram. The enthalpy of the B e y transition was determined by d.t.a. as 3.8+ 1.0 kJ mol-l.

The /?+y transition in complex oxides such as Li,ZnSiO,, Li,PO, and NaAlO, involves a redistribution of the cations over the available tetrahedral sites in an essentially unchanged, approximately hexagonal close packed, oxide framework.' In Part 1, a quantitative powder X-ray diffraction method was developed for measuring the kinetics of the transition in Li,ZnSiO, and it was shown that the data may be analysed using the Avrami equation., Crystallographic aspects of the transition were also discussed in Part 1. In Part 2, kinetic data are presented for the /?+ y transition in Li,ZnSiO, in both forward and reverse directions. The influence of temperature and particle size on the kinetics is discussed and a model presented for the temperature dependence of the transition rates.

RESULTS The kinetics of the transition P + y Li,ZnSiO, were followed by using X-ray

powder diffraction to measure the relative amounts of the /? and y polymorphs, as described in Part 1., The equilibrium transition temperature between the low- temperature /? polymorph and the high-temperature y polymorph is estimated as 880 f 20 OC. The transition y --+ /? was studied isothermally at eleven temperatures in t Permanent address: Instituto de Investigaciones en Materiales, U.N.A.M., Mexico 20 DF, Mexico.

2297

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2298 K I N E T I C S OF P O L Y M O R P H I C T R A N S I T I O N S

the range 480-855 "C. The reverse transition, B + y , was studied at three temperatures and over a more limited range, 906-940 O C .

The results are shown as reduced time plots fig. 1 , in which the fraction transformed is plotted against time. The data for 772 and 803 O C , which were very similar, are shown in fig. l(a) and (b) in order to make comparison between the different temperatures easier. The transformation rate was slowest at 480 O C ; the rate became more rapid with increasing temperature and passed through a maximum at 772 and 803 O C . Above 830 O C , the rate decreased very rapidly and at temperatures in the range ca. 860-900 OC no detectable transformation occurred in either direction for times of up to two weeks. Above 906 OC the rate again increased (but now in the direction /3 + y ) with increasing temperature, and above ca. 940 O C the kinetics could not be studied by the present techniques since the transformation was essentially complete within a few minutes.

0

I 1 I I I I I

1 0 10 0 100 0 t lh

t lh FIG. 1.-Reduced time plots for the transition y e p Li,ZnSiO,; temperatures ("C) are marked. 480 to

855 O C , direction y -, /I. 906 to 940 O C , direction B + y .

From inspection of fig. 1 it is clear that the data are not isokinetic : the curves cannot be superposed simply by translating them along the time axis. Apart from the data for 855, 480 and possibly 536 OC, the results do, however, fall into a general pattern. The data are approximately linear over the range ca. 30-90% transformation and the slopes show a gradual variation in which an increased slope appears to be associated

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M. E. V I L L A F U E R T E - C A S T R E J O N A N D A. R. WEST 2299

with an increased rate of reaction. This applies to the transition in both directions. Thus, the data at 940 "C for the direction /? + y fall between the data at 655 and 734 O C

for the direction y +/?, in terms of both the rate of reaction and the slope of fig. 1. At 480 and 855 "C, the reaction rates were the two slowest rates at which data were recorded and the nature of these results in fig. 1 is quite different from the other results.

JOHNSON-MEHL-AVRAMI ( J M A ) A N A L Y S I S

In Part 1 it was shown that the data at most temperatures (but not 855 "C) may be described by the Avrami equation.2 This equation, also called the Johnson-Mehl equation, is of the form:3

!!? = k n t n - 1 ( 1 - y ) ( 1 ) dt

where y is the fraction transformed. If k and n are constants for a particular reaction the equation may be rewritten as:

y = 1 -exp ( -k t )" . (2) If this equation is obeyed, then a plot of log log[ 1 / ( I - y ) ] against log time should give a straight line of slope n and intercept n log k-log e. Admittedly, such a plot is insensitive, but it does serve to give a value for the exponent n. The data were replotted in this form and straight lines obtained from all temperatures, apart from 480 and 855 OC. A typical plot was shown in Part 1, fig. 9.2 Values of n are listed in table 1 ; an approximate correlation exists between the magnitude of n and the slopes of the lines in fig. 1.

TABLE 1 .--KINETIC DATA FOR THE TRANSITION /?$y Li,ZnSiO,

T/OC lo3 K I T n k/h--l

480 536 570 602 655 734 745 772 803 839 855 906 922 940

1.328 1.236 1.186 1.143 1.078 0.993 0.982 0.957 0.929 0.899 0.887 0.848 0.837 0.824

not const. 0.54 0.55 0.61 0.70 1.02 1.15 0.98 1.08 0.75

not const. 0.50 0.56 0.6 1

(0.0051) 0.073 0.174 0.692 1.18 4.17 5.13 7.08 7.08 0.89

(0 -0 1 4) 0.024 0.371 2.34

The values of k were obtained directly from fig. 1. Assuming that eqn (2) holds. k-l is equal to the time at which

e- 1 e

y = - = 0.632. (3)

These data are also given in table 1 ; values for 480 and 855 OC are included, although there must be uncertainty about their significance when compared with the values for other temperatures since the data at 480 and 855 O C do not fit the JMA equation.

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2300 KINETICS O F P O L Y M O R P H I C T R A N S I T I O N S

The JMA equation is an empirical equation which describes the overall kinetics of a transformation. It does not attempt to separate the nucleation and growth stages although it has been suggested that information about the transformation mechanism may be obtained from the n value.4 In the present work, the JMA equation is used simply as an empirical equation which is found to fit most of the data reasonably well. The main aspect of this work is regarded to be the temperature dependence of transformation rates, rather than the precise mechanism, which appears to be complex and may vary with temperature.

A U T O C A T A L Y T I C KINETICS

The data for 480 and 855 O C do not fit the JMA equation. The sigmoidal shape of their curves, fig. 1 , suggests that the transformation may be autocatalytic at these temperatures. This would occur if the rate is a function of the concentrations of both B and y phases. The simplest autocatalytic case is for a reaction that is first order with respect to the reactant, in which case the rate is given by:3

9 = ky(1 - y ) dt (4)

or ( 5 )

This equation can only apply to the later stages of a reaction since it is indeterminate for y = 0, t = 0.

- 3.0 b I I I I I I I I I 1 I I I

0 20 40 60 80 100 120 140 160 180 200 220 240 260

FIG. 2.-Plots which indicate autocatalytic behaviour. t lh

The applicability of eqn (5) to the 480 and 855 O C data was tested by plotting In [ y / ( 1 - y)] = In ( P l y ) against time, fig. 2. The data fall on straight lines, apart from the early stages of transformation, and show that at these temperatures the transformation may indeed be autocatalytic. Data for 536 and 574 O C show some evidence of linearity during the later stages but for all other temperatures, e.g. 839 OC, fig. 3 , the data are curved and the transformation is clearly not autocatalytic.

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M. E. V I L L A F U E R T E - C A S T R E J O N A N D A . R . W E S T 230 1

1 0 *

- 2 o t 1 I I I I I I

10 2 0 t lh

FIG. 3.-Plot which does not indicate autocatalytic behaviour.

1000-

- u 0 800- L

600 -

FIG. 4.-Time-temperature-transformation diagram for the transition B < y Li,ZnSiO,. Curves are for 25, 50 and 75% conversion. Data points are shown for 75% only.

T I ME-TE M P E R A TU RE-T R A N S F 0 R M A T I 0 N (TTT) D I A G R A M S

It is common practice, especially amongst metallurgist^,^ to plot kinetic data for phase transitions as TTT diagrams. The time taken to achieve a certain degree of conversion, say 25%, is ascertained over a range of temperatures and a graph of temperature against log time made. A family of curves for different degrees of conversion may then be constructed. Results are shown in fig. 4 for the p y transition in Li,ZnSiO, for 25, 50 and 75% conversion; the resulting curves are similar to those observed in metals? Transformation rates are essentially zero for a range of temperatures to either side of the equilibrium transformation temperature, 880 O C . At lower temperatures, the time needed to achieve a certain degree of conversion passes through a minimum at ca. 790 O C .

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2302 KINETICS OF P O L Y M O R P H I C T R A N S I T I O N S

T E M P E R A T U R E D E P E N D E N C E A N D A R R H E N I U S A N A L Y S I S

The temperature dependence of the transition rates is shown in fig. 5 as an Arrhenius plot of log k against reciprocal temperature. The values of k were obtained from the JMA analysis, table 1. For those temperatures at which the data could be analysed by the autocatalytic equation (480 and 855 "C) it is possible to obtain an alternative rate constant value. However, since the rate constants obtained from the two analyses are not equivalent, the autocatalytic values are not included in fig. 5.

Several regions on the Arrhenius graph, fig. 5 , may be distinguished. Below ca. 770 O C the graph appears to be linear for the transformation direction y + p; from the slope, an activation energy of 147 20 kJ mol-l is obtained. The datum point for 48OOC lies off the line; however, the JMA equation is not applicable at this temperature and hence there is large uncertainty over this datum point. Between ca. 800 and 855 OC the rate of the y -+ fi transition decreases increasingly rapidly. Between ca. 860 and 900 OC the rate could not be measured in the time available. Above ca. 900 O C the fi -+ y transition takes place increasingly rapidly and above ca. 950 O C the transition is complete within a few minutes. This placed an upper limit on the transition rates that could be studied experimentally since the samples took a few minutes to equilibrate at the temperature of the furnace. Insufficient data were available to determine an activation energy for the transformation in the p + y direction.

EFFECT OF P A R T I C L E S I Z E O N THE RATES OF T R A N S I T I O N

All the results in table 1 and fig. 1 were obtained on material from the same batch which, hopefully, had the same distribution of particle size throughout. The material came from a sample of Li,ZnSiO, which had been sintered at ca. 1200 O C for one day and then crushed to a fine powder in an agate mortar. The sample was not sieved and particle size analysis was not carried out but, from a cursory examination with a polarising microscope, most particles appeared to have sizes in the range 1-30 pm with a few particles as large as ca. 50 pm.

The effect of particle size on the y + f i transition rate was assessed by preparing the following samples and studying their transition rates at one temperature, ca. 572 OC: Sample A, control sample taken from the above batch. Sample B, a sample from the above batch which was subjected to prolonged grinding. From a microscopic examination, most of the grains were smaller than ca. 20 pm diameter. A lower limit to the grain size could not be fixed but must have been between ca. 1 (the lower limit of resolution of the microscope) and 0.1 pm (X-ray powder diffraction line broadening would be expected to occur for crystal sizes < ca. 0.1 pm and this was not observed). Sample C, a sample of Li,ZnSiO, was sintered at a higher temperature, ca. 1350 OC, in order to increase the crystal size by grain growth. The resulting sinter was crushed only lightly and crystals of up to ca. 100 pm diameter were found to be present. Sample D, a sample from the standard Li,ZnSiO, batch, was converted to the p polymorphs by heating overnight at ca. 700 *C and then converted back to the y form by heating at 1100 OC for a few hours. Since volume changes accompany the fie y transition it was thought that these may generate stresses within the crystal or even lead to fragmentation of crystals and a reduction in particle size.

The results for samples A to D are given in fig. 6 as reduced time plots. Qualitatively, it is clear that the rate of transition increases with decreasing particle size: the rates decrease in the order B > A > C . The value of n also varies with particle size but since the data show some scatter, this variation may not be significant. The variation of transition rate with particle size suggests that nucleation at the crystal surface may be the rate controlling step but further information is needed to confirm this. Perhaps

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M. E. V I L L A F U E R T E - C A S T R E J O N A N D A, R. W E S T

TIo C 7000 800 600 500 I I I 1 I 1

Tc = 880!20"C

2303

lo3 KIT FIG. 5.--Temperature dependence of the rate constant for the transition B e y Li,ZnSiO, in Arrhenius

format .

0.1 1.0 tlh

10

FIG. 6.-Effect of particle size on the kinetics of the y -.B transition. A, normal sample. B, finely ground sample. C, sample with larger grain size. D, sample previous cycled through y + B -+ y. For each

sample, temperatures were constant to within & 2 "C.

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2304 K I N E T I C S O F P O L Y M O R P H I C T R A N S I T I O N S

the most surprising result is that for sample D, which had the slowest transition rate. This showed that the effect of previously cycling the sample through the sequence y -+ P + y was to decrease the rate of the y -+ P conversion on the second cycle.

EFFECT OF OTHER PHASE T R A N S I T I O N S O N THE KINETICS

The P and y structures of Li,ZnSiO, are, in fact, two families of structures since and y phases exhibit polym~rphism.~ For both families there is considerable both

thermal activity in the temperature range ca. 620-670 "C due to the transitions:

Y o -+ YI + 711

The details of the structural changes that occur during these various transitions are not known but they probably involve small distortions or rotations of the MO, tetrahedra without the necessity of breaking and reforming any primary bonds. These internal polymorphic changes within the p and y families of structures appear to have no influence on the rate of the y -+ P transition since the data at 602 and 655 O C , table 1 and fig. 1, do not in any way appear to be anomalous. HedvalP has reported many examples of increased reactivity of phases at temperatures in the vicinity of a phase transition, e.g. AgI at ca. 145 O C , but no evidence for the occurrence of such an effect was seen in the present study.

PI + PI. + P I V + P I P

+ - 3 4

: n m

3 3

KCI

- NaCl

600 700 800 T/"C

FIG. 7.-Calibration of d.t.a. method using known fusion enthalpies.

exo

t J-

AT

endo

200 400 600 800 1000

Tl" C FIG. 8.-D.t.a. of B,Li,ZnSiO, heated at 20 OC min-'.

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M. E. V I L L A F U E R T E - C A S T R E J O N A N D A. R. W E S T

T H E R M O D Y N A M I C S OF T H E T R A N S I T I O N P + y Li,ZnSiO, The enthalpy of theB+ y transition was determined approximately using differential

thermal analysis (d.t.a.). The instrument used (Du Pont 900, 1200 O C cell) is designed for high sensitivity rather than calorimetric response and hence the enthalpy value is only approximate. The method was calibrated using materials with known transition enthalpies. Alkali halides were used with melting points in the range 450-800 OC. For a given sensitivity of the instrument, the ratio (peak area: enthalpy of fusion) was approximately constant, fig. 7. A d.t.a. trace ofp,Li,ZnSiO, on heating at 20 OC min-l is shown in fig. 8. Two endothermic peaks are observed, one at ca. 650 O C due to the transition Pi + /IIi and the other at ca. 940 O C due to the BI1 y.yII transition. The latter temperature is considerably above the equilibrium transition temperature, ca. 880 OC, and is further evidence that the transition exhibits considerable hysteresis and takes place only slowly at temperatures close to T,. From the area of the Prr -+ yII. peak, a transition enthalpy of 3.8 1 .O kJ mol-1 was determined from fig. 7 assuming the formula to be Li,ZnSiO, with a formula weight of 171.

2305

DISCUSSION

Two methods of presenting the temperature dependence of the transition rate data have been used. The Arrhenius plot is the method conventionally used in kinetic studies on inorganic compounds, whereas for phase transitions in metals and aIloys it is more common to use TTT diagrams. The TTT method has the advantage that no assumptions about transition mechanism and rate equations are needed whereas attention to such matters is necessary to construct an Arrhenius plot of rate constant data. The TTT diagram presented here shows many similarities to TTT diagrams in alloy systems and suggests thateth y could be applied generally to transformations in non-metallic systems.

An unusual feature of the present study is the wide range of temperatures over which the kinetics have been studied (ca. 460 "C) together with the study of the transition in both forward and back directions. In order to obtain an activation energy from an Arrhenius plot for the present transformation, it is necessary to avoid temperatures close to T, because of departures from linearity: departures from Arrhenius behaviour occur for temperatures of up to at least ca. 100 O C below T, and for at least 60 O C above c. It has not been possible, therefore, to obtain an activation energy for the direction B -+ y because of insufficient data.

At temperatures close to T,, the difference in free energy between the p and y polymorphs is small. The explanation usually offered for the reduction in transition rates close to T, is that nucleation is the rate controlling step and is difficult. In order for a nucleus of the product phase to be stable, the reduction in free energy on formation of the product must be greater than the positive surface energy of the nucleus. This leads on to the idea of a critical size for nuclei, below which they are unstable. As T, is approached, the change in free energy accompanying the transition decreases and the critical size of the nuclei increases; hence nucleation becomes more difficult. In practice, however, it is usually difficult to make quantitative assessments of nucleation rates and theoretical calculations are hampered because the surface energies of nuclei are not usually known.

We investigate here the possibility of an alternative approach to the temperature dependence of transformation rates which is independent of the mechanistic details. For the transition y , at any temperature T ( T # T,), the transition must eventually proceed to completion in one direction since, from the phase rule, the two phases B and y may coexist in equilibrium at only one temperature, T,. Hence, there must be

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2306 K I N E T I C S OF P O L Y M O R P H I C T R A N S I T I O N S

a net, macroscopic transformation in one direction. However, let us suppose that, microscopically, both forward and back reactions may occur, at different rates, and that it is the difference between these microscopic rates that governs the net transition rate and direction. One example of how this might arise is as follows. Suppose that the transformation occurred by growth of crystals of one phase, say p, at the expense of the other, y, such that the rate of transformation was controlled by the movement of the interface between ,4 and y phases. Individual atoms may hop, reversibly, across the interface but for any net movement of the interface to occur the concerted hopping of many atoms, at the interface, would be necessary. Hence the rate of movement of the interface would be associated with the difference in hopping rates in the two directions.

Let us suppose that the microscopic hops in the two directions may be represented by two rate equations:

and

The overall rate constant, k , is given by the net difference between the two individual rate constants, i.e.

If the activation energies and prefactors are known, k may be calculated as a function of temperature. At the transition temperature, let kP, , = k,,B and hence k = 0. However, it may be necessary to distinguish two transition temperatures. One corresponds to the macroscopic free energies of the two phases and is estimated as 880 f 20 O C for the p+ y transition. The other is relevant to the microscopic processes which occur at the P l y interface; for the direction y -+ p, let this temperature be ca. 860 O C since, from fig. 5, the rate of the y -+p transformation drops to zero at about this temperature.

Using the experimental results for E,,B and A,,P, values of Ep,, and Ap,r may be suggested that satisfy the condition k = 0 at 860 O C . In fig. 9 are shown the experimental rate constant data and those calculated for the direction y -+ p, assuming that Ep, = 167 kJ mol-l. By adjusting the two prefactors, and the calculated curve may be translated vertically and brought into superposition with the experimental curve. Similar curves may be generated for other values of E , and it is observed that the curve is fairly insensitive to small changes in EP,y. The cioice of 167 kJ mol-l for EP,, is therefore not critical.

The closeness of the fit between experimental and calculated curves, fig. 9, does not, of course, prove the correctness of the model for the temperature dependence of the y -+ p transition rate. However, it does justify its serious consideration, especially since the model may be applicable to first-order phase transitions in general.

kj?, y = Ap, y exp (- Ep, y I W k,,p = A,$ exp (-E,,p/RT)*

= k,,P-kP,Y

C O N C L U S I O N S The rate of the transition j?e y Li,ZnSiO, may be described at most temperatures

using the Johnson-Mehl-Avrami equation. Rates are very dependent on particle size and increase with prolonged grinding of the sample prior to transformation.

By using powdered samples from the same source for each experiment, it was possible to assess quantitatively the temperature dependence of the kinetics. At temperatures well-removed from the equilibrium transformation temperature, the rate constant for the y - + P transformation fits the Arrhenius equation with an activation energy of 147 20 kJ mol-l. However, at temperatures closer to T, the rate constant decreases increasingly rapidly.

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M. E. V I L L A F U E R T E - C A S T R E J O N A N D A. R. W E S T 2307

0°- I .y I

-1 0 - I 1 I ' 3 0"

\

1 I I 1 1 1 I 1

0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 1 5

FIG. 9.--Comparison between experimental (---) and calculated (-) rate constant data. The calculated curve may be translated up or down by adjusting the pre-exponential factors.

In the range ca. 860-900°C, no net transformation could be detected, in either direction, for heating times of up to two weeks. Possibly, strain energy associated with the P l y interface is important at these temperatures; thus, the orthorhombic unit cells of the p and y polymorphs have similar a and b dimensions but c is a few percent larger in y.5 Hence the volume change associated with the transformation may generate stresses which modify the free energies of the phases involved.

The temperature dependence of the y -+ P transformation rate has been modelled satisfactorily by regarding the transformation as a modified, reversible reaction at all temperatures. This approach takes the idea of microscopic reversibility, such as is applied to gas-phase reactions, and imposes the constraint that the phase rule holds and therefore, in a one-component system like Li,ZnSiO,, two phases may coexist in equilibrium at only one temperature. The apparent success of this approach suggests that it may be more widely applicable.

103 KIT

A.R.W. thanks the S.R.C. for a research grant and P. G. Bruce for fruitful discussions.

A. R. West, Z . Kristallogr., 1975, 141, 422 and references therein.

J. Burke, The Kinetics of Phase Transformations in Metals (Pergamon Press, New York, 1965). C. N. R. Rao and K. J. Rao, Phase Transitions in Solids (McGraw-Hill, New York, 1978) and references therein. A. R. West and F. P. Glasser, J . Muter. Sci., 1970, 5, 557, 676. J. A. Hedvall, SoIid State Chemistry (Elsevier, Amsterdam, 1966).

* M. E. Villafuerte-Castrejon and A. R. West, J. Chem. Soc., Faraday Trans. I , 1979, 75, 374.

(PAPER 91 1857)

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