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A model for non-isothermal crystallization kinetics
]OIJRNA£ OF
ELSEVIER
Journal of Non-Crystalline Solids 208 (1996) 288-293
A model for non-isothermal crystallization kinetics V. Erukhimovitch, J. Baram * Materials Engineering Department, Ben-Gurion University, P.O.Box 653, 84105 Beer-Sheva, Israel Received 25 January 1996; revised 7 May 1996
Abstract
In the framework of a previously proposed Integral Equation formalism, an equation is derived for the kinetics of non-isothermal crystallization (devitrification). It is shown that the Kissinger multiple-scan thermal analysis technique, commonly used for the evaluation of activation energies, cannot be applied to all non-isothermal transformations, but only to those where site saturation, heterogeneous nucleation takes place. The heating rate is the controlling factor of the nucleation mode in devitrification. If it is high enough, homogeneous nucleation is activated. Heterogeneous nucleation takes place when the heating rate is low. In that case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization. A critical heating rate is derived, that depends on the amount of pre-existing nucleation sites, on the activation energy for growth of the new phase and on a critical temperature. The new model is conform to the results obtained in devitrification experiments induced by high rate pulse heating, as compared to those obtained at lower heating rates usually practiced in DSC analyses for the same material system.
1. Introduction
When the kinetics of nucleation and growth transformations are determined experimentally, by thermal analysis or other techniques, at several different constant temperatures, a complete isothermal t i m e temperature-transformation ( T I T ) diagram can be drawn [1]. Such a diagram gives the relation between the temperature and the time for fixed fractional amounts of transformation to be attained, and can therefore be used to evaluate the quenching rates necessary for obtaining the product phase of the transformation [2]. Isothermal curves have been widely interpreted in the past in terms of the Kol-
* Corresponding author. Tel.: +972-7646 1474; fax: +9727647 2944; e-mail: [email protected].
m o g o r o v - J o h n s o n - M e h l - A v r a m i (KJMA) formulation [3-5] for the transformation kinetics. Popular thermal analysis interpretation methods (Kissinger [6,7] and Ozawa [8,9]) are frequently used to calculated the thermodynamic and kinetic parameters of the transformation resulting from experiments performed at different heating rates, i.e., in non-isothermal conditions. Henderson [10] has provided a theoretical basis for the treatment of non-isothermal thermal analysis techniques and has shown that the Kissinger relation for evaluating the activation energy, though originally based on the order of reactions, can also be derived from the KJMA formalism [10,11]. It has been shown recently that for transformations that proceed isothermally, the generally accepted KJMA formalism is only a simplification of the real physical conditions for all cases where nucleation proceeds as a continuous process [12-15].
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 ( 9 6 ) 0 0 5 2 1 - 2
V. Erukhimovitch,J. Baram/ Journalof Non-CrystallineSolids208 (1996)288-293 Fitting experimental data to the KJMA equation is often reported to cause an overestimation of the transformed volume, probably resulting in incorrect evaluations of the kinetic and thermodynamic parameters for the studied isothermal transformation. By using the same physical assumptions that guided original Avrami's development [4], integral equations were derived [12-15] for a more accurate expression of the isothermal transformations kinetics, for either homogeneous or heterogeneous nucleation (with constant or varying nucleation rates) and interface- or diffusion-controlled growth isothermal transformations. Ref. [14] presents an extensive treatment of the inadequacies of the KJMA formalism, and shows the corrections introduced by the new Integral Equation model. Lately, this formalism has also been applied to primary recrystallization that may occur in work-hardened metals or alloys [15]. It is the purpose of this paper to adapt the new formulation to non-isothermal transformations, and to present the derivation of a critical heating rate in non-isothermal devitrification experiments. 2. Model formulation Like Avrami's treatment [4], this paper and the previous ones [12-15], are based on the following physical and mathematical assumptions: (A1) nucleation is a uniform random process; active nucleation sites for crystallization are randomly distributed throughout the volume; (A2) as time proceeds during growth, some product volumes grown from sites active at various times in the past, impinge upon product volumes grown from other sites. Growth is terminated at points (or surfaces) of mutual contact, though it continues normally elsewhere. If considered rigorously, the randomness condition (A1) for the active nucleation sites must apply to the untransformed volume only [12-15], thus eliminating the need for Avrami's 'phantom nuclei' concept [4]. These basic assumptions are believed to hold in both the isothermal and the non-isothermal cases. It should
i This additional condition is explicitly mentioned in Ref. [16], where the Kolmogorov model is presented. It is however unfortunate that in the same paper [16], a confusing notation is used: Eq. (1) includes the symbol V for the non-crystallized volume, while Eq. (3) uses the same symbol for the transformed volume.
289
however be stressed that for these assumptions to hold in both cases, crystallization should take place in a non-limited medium, so that boundary terms are ignored [16,17]. Moreover, surface nucleation is not considered either, as the transformation kinetics in that case have been shown not to conform to the KJMA formulation [18,19].
2.1. Non-isothermal crystallization For a non-isothermal nucleation and three-dimensional, interface-controlled growth transformation 2, that is characteristic for melting or solidification from the liquid state, the transformed volume issued from one single active nucleation site at time t and temperature, T, is given by
IJz
4"rr z(r, t) = T
tG(T, ~')dT
1
(1)
where z is the nucleation time, and G(T, z) is the temperature and time-dependent growth rate of a single nucleus of the product phase. Within the untransformed volume, V~(T, t)= Vtota1 - V~(T, t), the number of new nuclei formed in the time interval z and z + d z is
d U = t ( T , z)[Vtota,-VO(T, Z)] dz, (2) where I(T, z) is the temperature and time-dependent nucleation rate. Mutual interference of grains growing from different nuclei causes growth to cease in regions with common interfaces, when the condition V t3 << V '~ is not satisfied anymore. The problem is primarily geometrical, and has been treated by Avrami [4] by introducing the concept of 'extended volume' of the transformed material, Vex~t,which, in the non-isothermal case is given by V~x~tended(T, t) = =
f£dU(z) ~3(T, t) fotI(T, Z)[Vtota ' --
k
x ~G(r, z ) d r
V f l ( T , 7)]
1
dz
(3)
e Treated here as an example, without loss of generality, though crystallization from the amorphous state is often a diffusion controlled growth transformation.
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
290
ation rate, i.e., in the site-saturation case, Eq. (6) reduces to
and one can write d Vex~tended
dz
¢v(T,
t)= 1-exp
[--~-No[JoC(T,
r)d~
,
(8) (4) The relation between Ve~t(T, t) and V¢(T, t), which is meant to correct for the geometric problem of impingement, has been given by Avrami [4] and later adopted by Christian [1], as 3
dVI3(T, Z)
dVflxt(T, z)
1
V~(T, Z)
Vtota~(r, z)
=l-~v(T,
z)
(5) and therefore one gets the following integral equation for the description of the kinetics 4
~v(r, t) =
4'rr ct -~-Jolv(T, z) [fz tG(T, r ) d r ]3 × [1 -
~v(r,
z)] 2 dz
(6)
rather than the non-isothermal KJMA equation [1]
¢v(T,
47r ~t
t) = 1 - e x p
---~-Jol(T, z)
x[fztG(T , T) d'r]3 d z l .
(7)
When the interface-controlled, 3D growth nonisothermal transformation proceeds at zero nucle-
3 The relationship between the real transformed volume and the extended volume has been established for isotropically growing spheres of the product phase. However, Price [20] has shown that computer simulations do demonstrate that the extended volume concept provides 'reasonable compensation for grain impingement' [20] in various geometries (spheres, bipyramids, ellipsoids) and for various aspect ratios [20]. The cases treated by Price were for recrystallization with site saturation (zero nucleation rate), where 'phantoms' are excluded in the first place. 4 Similar non-isothermal kinetic equations can be derived for diffusion-controlled transformations, and for growth in 3, 2 or 1 dimensions.
where NO is the initial number of nucleation sites, which is the classical KJMA transformation rate equation [ 1]. Eq. (6) is not a true kinetic equation. The rate of evolution of the product phase cannot be expressed as a product of two functions, one of temperature, the other of the fraction transformed [14]. The use of the Kissinger or Ozawa methods [6-9] for the evaluation of the thermodynamic and kinetic parameters of the transformation, is therefore not appropriate in the cases where Eq. (6) prevails. However, the Kissinger or Ozawa methods are applicable in the case of site saturation, i.e., the case where Eq. (8) is the one that reflects the kinetics of the non-isothermal transformation.
2.2. Critical heating rate in devitrification In an early paper [21] on non-isothermal crystallization of rapidly heated glasses, it has been reported that "activation energies calculated using the Kissinger method differ significantly for rapid and slow heating rates .... indicating different crystallization mechanisms". The material was a FeNiSiB glass, and the heating rates w e r e 10 6 K s -l , by pulse heating, and about 102 K s -1, by differential scanning calorimetry [21]. The study of the nucleation and growth of stable and metastable eutectics in FeSiB glasses [22] has shown that the stable eutectic cell size distribution is different when the material (Fe72Si10B18) is annealed at either 723 or 793 K, due to the change in the modes of nucleation, from heterogeneous, site saturation, (at the lower temperature) to homogeneous (at the higher temperature). Another observation related to isothermal crystallization of undercooled metallic melts and metallic glasses around the glass transition temperature [23], mentions that "...above Tg, the atomic transport is controlled by viscous flow rather than by diffusion as at temperatures below the glass transition .... thus leading to a very large increase in the nucleation rate above Tg". Moreover, according to Ref. [23],
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
metal-metalloid (Fe66Ni10B24) and transition metal (ZrsoCos0) glasses both crystallize by homogeneous nucleation above Tg, and by heterogeneous nucleation below Tg. Clearly, both nucleation modes have different activation energies. It can be inferred that if the heating rate in devitrification of a specific metallic system is high enough, the nucleation might be activated homogeneously. Conversely, heterogeneous nucleation will take place in that system when the heating rate is low. In that case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization, which itself has been induced by heterogeneous nucleation. If, as reported in the literature, the data analyses of many devitrification experiments [20] show serious deviations from the ' A r r h e n i u s ' behavior expected from the Kissinger/Avrami theory, crystallization has probably taken place by homogeneous nucleation, and probably at a large heating rate. An evident conclusion of this duality is the existence of a critical heating rate in non-isothermal crystallization processes. If the heating rate is below the critical rate, heterogeneous, site saturation, nucleation prevails. Conversely, if the heating rate is above that critical rate, the operative mode is homogeneous nucleation. The integral equation formalism [12-14] has shown that the KJMA rate equation is still valid in the case of heterogeneous nucleation taking place at randomly dispersed pre-existing nuclei, i.e., in the case referred to as 'site saturation'. It follows that the Kissinger multiple-scan technique and peak method [6,7], which conforms to the KJMA rate equation for assessing the kinetics in non-isothermal devitrification experiments [10], is applicable only to that specific case, very often encountered indeed in non-isothermal devitrification experiments. The limiting, critical heating rate can therefore be derived from the KJMA equation, which is the adequate formalism for heterogeneous (site saturation) nucleation. Let's define: ]b= (dT/dt). Eq. (8) may then be written as
~v(T,
t) = 1 - e x p [
4~ No[frG(T,)dT,] 3] 3 ~,3 [To J (9)
291
The temperature dependence of the growth rate is an Arrhenius type function of the activation energy, E c, for growth only of the crystalline phase:
10, where G o is a material parameter. Integration of Eq. (10) in the temperature interval of the non-isothermal devitrification process gives
frG(T)dT=Gofrexp[@-]dT, To
To
(11)
[
which can be approximated [24] as
- ] dT~- G°EG 10-2"315-0'4577(EG/RT). Gofrexp[ -EG ~r,,
[ Rr
R
(12) The relation between the heating rate, 7~, and the crystallized volume fraction, ~v(t), can now be formulated, after some algebraic manipulations i? = 7.7 × 10 - 3
GoEc -
-
10 -0"4577(EG/RT)
R
×
-No ln[1 - ~(t)] "
(13)
It can be shown by a simple computation that if the volume fraction transformed following heterogeneous nucleation, for which Eq. (13) is applicable, ranges between 99.999% and 99.900%, the contribution of the term ~ - 1/ln[1 - ~(t)] to the value of ik is 0.48 _ 0.04, meaning that the heating rate does not change by more than 10%. In the case of almost complete heterogeneous devitrification ( ~ v ( t ) > 99.999%), a critical heating rate, i/'crit,can be approximated as follows: Lrit = 4.4 X
lO-4GoEalO-°°55(ea/rc,03~o.
(14)
Eq. (14) shows that the critical heating rate depends strongly on both the values of the activation energy, E G, for growth of the crystalline phase and of the critical temperature. T~rit has to be chosen as the temperature for which the density of the homogeneously nucleated sites, as derived from classical thermodynamics, is significantly higher than the density of the pre-existing, heterogeneously nucleated
292
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
sites N0. In fact, Tcrit is the highest temperature at which the heterogeneous, site saturation, nucleation mode is still the dominant one. An indicator that the transition from the site-saturation heterogeneous nucleation mode to the thermally activated homogeneous nucleation mode indeed occurs, is when the Avrami exponent derived from the kinetic analysis of an isothermal experiment changes to a value higher than 3 (the value predicted by the crystallization kinetics theory for 3D, interface-controlled), as reported in Ref. [22], and others mentioned in the same paper 5. Again, if the heating rate is below the critical rate, Tcrit, heterogeneous site saturation nucleation prevails. Conversely, if the heating rate is above Tcrit' the operative mode in the considered transformation is mainly homogeneous nucleation. Experimental support for this finding may be deduced from the results reported in Ref. [21], fig. 6. Two extreme heating rates (by pulse heating and by ordinary DSC experiments, practiced on the same material system) resulted in entirely different traces (slopes) of the computed Kissinger plots. Calculated activation energies were reported to differ for rapid and slow heating rates: 290 kJ mo1-1 and 140 kJ mol -~, respectively. According to the model presented in this paper, the DSC data reflects heterogeneous nucleation, with site saturation. The involved activation energy in this case is the one for growth only of the crystalline phase. Conversely, in view of the above discussion, the dependence of peak temperatures on heating rates in the case of the pulse heating experiments, which probably reflect an homogeneous nucleation and growth process, cannot be analyzed by the Kissinger method, applicable only to heterogeneously nucleated transformations.
3. Conclusions In the framework of the Integral Equation formalism, an equation is derived for the kinetics of nonisothermal crystallization (devitrification). When the operative nucleation mode is of the heterogeneous,
5 For 3D diffusion-controlled transformations, the corresponding Avrami exponent is 3/2.
site saturation type, the new formalism reduces to the KJMA equation. It is shown that, as a result, the Kissinger multiple-scan technique used for the evaluation of activation energies, is not applicable to all non-isothermal transformations, but only to those where site saturation heterogeneous nucleation takes place. In a devitrification transformation, the heating rate controls the nucleation mode. If the heating rate is large enough, homogeneous nucleation is activated. Conversely, heterogeneous nucleation takes place when the heating rate is small. In that last case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization, which has been induced by heterogeneous nucleation. A critical heating rate has been derived, that depends on the amount of pre-existing nucleation sites, on the activation energy for growth of the new phase and on a critical temperature, the temperature at which the density of the homogeneously nucleated sites is significantly higher than the density of the pre-existing, heterogeneously nucleated sites.
Acknowledgements One of the authors (V.E.) acknowledges financial support by the Israeli Ministry of Science and Arts during the period of this research.
References [1] J.W. Christian, The Theory of Transformations in Metals and Alloys, Part I Equilibrium and General Kinetic Theory (Pergamon, Oxford, 1975) section 59. [2] H.G. Jiang and J. Baram, Mater. Sci. Eng. A208 (1995) 232. [3] A.E. Kolmogorov, Akad. Nauk. SSSR IZV Ser. Mater. 1 (1937) 355. [4] M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. [5] W.A. Johnson and R.F. Mehl, Trans. Am. Inst. Min. Eng. 135 (1939) 416. [6] H.F. Kissinger, J. Res. Natl. Bur. Stand. 57 (1957) 217. [7] H.F. Kissinger, Anal. Chem. 29 (1957) 1702. [8] T. Ozawa, Bull. Chem. Soc. Jpn. 38 (1965) 1881. [9] T. Ozawa, J. Therm. Anal. 2 (1970) 301. [10] D.W. Henderson, J. Non-Cryst. Solids 30 (1979) 301. [l l] L.V. Meisel and P.J. Cotte, Acta Metall. 31 (1983) 1053.
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293 [12] V. Erukhimovitch and J. Baram, Phys. Rev. B50 (1994) 5854. [13] V. Erukhimovitch and J. Baram, Phys. Rev. B51 (1995) 622. [14] J. Baram and V. Erukhimovitch, Innovations Mater. Res. 1 (1995) 109. [15] V. Erukhimovitch and J. Baram, Mater. Sci. Eng. (1996) in press. [16] M.P. Shepilov and D.S. Baik, J. Non-Cryst. Solids 171 (1994) 141. [17] M.C. Weinberg and R. Kapral, J. Chem. Phys. 91 (1989) 7146.
[18] [19] [20] [21]
293
M.C. Weinberg, J. Non-Cryst. Solids 134 (1991) 116. M.C. Weinberg, J. Non-Cryst. Solids 142 (1992) 126. C.W. Price, Acta Metall. 38 (1990) 727. A. Zaluska, D. Zaluski, R. Petryak, P.G. Zielinski and H. Matyja, in: Rapidly Quenched Metals, ed. S. Steeb and H. Warlimont (Elsevier Science, Amsterdam, 1985) p. 235. [22] M.A. Gibson and G.W. Delamore, Acta Metall. Mater. 38 (1990) 2621. [23] U. Koster and J. Meinhardt, Mater. Sci. Eng. A178 (1994) 271. [24] C.D. Doyle, J. Appl. Polym. Sci. 6 (1962) 639.
ELSEVIER
Journal of Non-Crystalline Solids 208 (1996) 288-293
A model for non-isothermal crystallization kinetics V. Erukhimovitch, J. Baram * Materials Engineering Department, Ben-Gurion University, P.O.Box 653, 84105 Beer-Sheva, Israel Received 25 January 1996; revised 7 May 1996
Abstract
In the framework of a previously proposed Integral Equation formalism, an equation is derived for the kinetics of non-isothermal crystallization (devitrification). It is shown that the Kissinger multiple-scan thermal analysis technique, commonly used for the evaluation of activation energies, cannot be applied to all non-isothermal transformations, but only to those where site saturation, heterogeneous nucleation takes place. The heating rate is the controlling factor of the nucleation mode in devitrification. If it is high enough, homogeneous nucleation is activated. Heterogeneous nucleation takes place when the heating rate is low. In that case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization. A critical heating rate is derived, that depends on the amount of pre-existing nucleation sites, on the activation energy for growth of the new phase and on a critical temperature. The new model is conform to the results obtained in devitrification experiments induced by high rate pulse heating, as compared to those obtained at lower heating rates usually practiced in DSC analyses for the same material system.
1. Introduction
When the kinetics of nucleation and growth transformations are determined experimentally, by thermal analysis or other techniques, at several different constant temperatures, a complete isothermal t i m e temperature-transformation ( T I T ) diagram can be drawn [1]. Such a diagram gives the relation between the temperature and the time for fixed fractional amounts of transformation to be attained, and can therefore be used to evaluate the quenching rates necessary for obtaining the product phase of the transformation [2]. Isothermal curves have been widely interpreted in the past in terms of the Kol-
* Corresponding author. Tel.: +972-7646 1474; fax: +9727647 2944; e-mail: [email protected].
m o g o r o v - J o h n s o n - M e h l - A v r a m i (KJMA) formulation [3-5] for the transformation kinetics. Popular thermal analysis interpretation methods (Kissinger [6,7] and Ozawa [8,9]) are frequently used to calculated the thermodynamic and kinetic parameters of the transformation resulting from experiments performed at different heating rates, i.e., in non-isothermal conditions. Henderson [10] has provided a theoretical basis for the treatment of non-isothermal thermal analysis techniques and has shown that the Kissinger relation for evaluating the activation energy, though originally based on the order of reactions, can also be derived from the KJMA formalism [10,11]. It has been shown recently that for transformations that proceed isothermally, the generally accepted KJMA formalism is only a simplification of the real physical conditions for all cases where nucleation proceeds as a continuous process [12-15].
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 ( 9 6 ) 0 0 5 2 1 - 2
V. Erukhimovitch,J. Baram/ Journalof Non-CrystallineSolids208 (1996)288-293 Fitting experimental data to the KJMA equation is often reported to cause an overestimation of the transformed volume, probably resulting in incorrect evaluations of the kinetic and thermodynamic parameters for the studied isothermal transformation. By using the same physical assumptions that guided original Avrami's development [4], integral equations were derived [12-15] for a more accurate expression of the isothermal transformations kinetics, for either homogeneous or heterogeneous nucleation (with constant or varying nucleation rates) and interface- or diffusion-controlled growth isothermal transformations. Ref. [14] presents an extensive treatment of the inadequacies of the KJMA formalism, and shows the corrections introduced by the new Integral Equation model. Lately, this formalism has also been applied to primary recrystallization that may occur in work-hardened metals or alloys [15]. It is the purpose of this paper to adapt the new formulation to non-isothermal transformations, and to present the derivation of a critical heating rate in non-isothermal devitrification experiments. 2. Model formulation Like Avrami's treatment [4], this paper and the previous ones [12-15], are based on the following physical and mathematical assumptions: (A1) nucleation is a uniform random process; active nucleation sites for crystallization are randomly distributed throughout the volume; (A2) as time proceeds during growth, some product volumes grown from sites active at various times in the past, impinge upon product volumes grown from other sites. Growth is terminated at points (or surfaces) of mutual contact, though it continues normally elsewhere. If considered rigorously, the randomness condition (A1) for the active nucleation sites must apply to the untransformed volume only [12-15], thus eliminating the need for Avrami's 'phantom nuclei' concept [4]. These basic assumptions are believed to hold in both the isothermal and the non-isothermal cases. It should
i This additional condition is explicitly mentioned in Ref. [16], where the Kolmogorov model is presented. It is however unfortunate that in the same paper [16], a confusing notation is used: Eq. (1) includes the symbol V for the non-crystallized volume, while Eq. (3) uses the same symbol for the transformed volume.
289
however be stressed that for these assumptions to hold in both cases, crystallization should take place in a non-limited medium, so that boundary terms are ignored [16,17]. Moreover, surface nucleation is not considered either, as the transformation kinetics in that case have been shown not to conform to the KJMA formulation [18,19].
2.1. Non-isothermal crystallization For a non-isothermal nucleation and three-dimensional, interface-controlled growth transformation 2, that is characteristic for melting or solidification from the liquid state, the transformed volume issued from one single active nucleation site at time t and temperature, T, is given by
IJz
4"rr z(r, t) = T
tG(T, ~')dT
1
(1)
where z is the nucleation time, and G(T, z) is the temperature and time-dependent growth rate of a single nucleus of the product phase. Within the untransformed volume, V~(T, t)= Vtota1 - V~(T, t), the number of new nuclei formed in the time interval z and z + d z is
d U = t ( T , z)[Vtota,-VO(T, Z)] dz, (2) where I(T, z) is the temperature and time-dependent nucleation rate. Mutual interference of grains growing from different nuclei causes growth to cease in regions with common interfaces, when the condition V t3 << V '~ is not satisfied anymore. The problem is primarily geometrical, and has been treated by Avrami [4] by introducing the concept of 'extended volume' of the transformed material, Vex~t,which, in the non-isothermal case is given by V~x~tended(T, t) = =
f£dU(z) ~3(T, t) fotI(T, Z)[Vtota ' --
k
x ~G(r, z ) d r
V f l ( T , 7)]
1
dz
(3)
e Treated here as an example, without loss of generality, though crystallization from the amorphous state is often a diffusion controlled growth transformation.
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
290
ation rate, i.e., in the site-saturation case, Eq. (6) reduces to
and one can write d Vex~tended
dz
¢v(T,
t)= 1-exp
[--~-No[JoC(T,
r)d~
,
(8) (4) The relation between Ve~t(T, t) and V¢(T, t), which is meant to correct for the geometric problem of impingement, has been given by Avrami [4] and later adopted by Christian [1], as 3
dVI3(T, Z)
dVflxt(T, z)
1
V~(T, Z)
Vtota~(r, z)
=l-~v(T,
z)
(5) and therefore one gets the following integral equation for the description of the kinetics 4
~v(r, t) =
4'rr ct -~-Jolv(T, z) [fz tG(T, r ) d r ]3 × [1 -
~v(r,
z)] 2 dz
(6)
rather than the non-isothermal KJMA equation [1]
¢v(T,
47r ~t
t) = 1 - e x p
---~-Jol(T, z)
x[fztG(T , T) d'r]3 d z l .
(7)
When the interface-controlled, 3D growth nonisothermal transformation proceeds at zero nucle-
3 The relationship between the real transformed volume and the extended volume has been established for isotropically growing spheres of the product phase. However, Price [20] has shown that computer simulations do demonstrate that the extended volume concept provides 'reasonable compensation for grain impingement' [20] in various geometries (spheres, bipyramids, ellipsoids) and for various aspect ratios [20]. The cases treated by Price were for recrystallization with site saturation (zero nucleation rate), where 'phantoms' are excluded in the first place. 4 Similar non-isothermal kinetic equations can be derived for diffusion-controlled transformations, and for growth in 3, 2 or 1 dimensions.
where NO is the initial number of nucleation sites, which is the classical KJMA transformation rate equation [ 1]. Eq. (6) is not a true kinetic equation. The rate of evolution of the product phase cannot be expressed as a product of two functions, one of temperature, the other of the fraction transformed [14]. The use of the Kissinger or Ozawa methods [6-9] for the evaluation of the thermodynamic and kinetic parameters of the transformation, is therefore not appropriate in the cases where Eq. (6) prevails. However, the Kissinger or Ozawa methods are applicable in the case of site saturation, i.e., the case where Eq. (8) is the one that reflects the kinetics of the non-isothermal transformation.
2.2. Critical heating rate in devitrification In an early paper [21] on non-isothermal crystallization of rapidly heated glasses, it has been reported that "activation energies calculated using the Kissinger method differ significantly for rapid and slow heating rates .... indicating different crystallization mechanisms". The material was a FeNiSiB glass, and the heating rates w e r e 10 6 K s -l , by pulse heating, and about 102 K s -1, by differential scanning calorimetry [21]. The study of the nucleation and growth of stable and metastable eutectics in FeSiB glasses [22] has shown that the stable eutectic cell size distribution is different when the material (Fe72Si10B18) is annealed at either 723 or 793 K, due to the change in the modes of nucleation, from heterogeneous, site saturation, (at the lower temperature) to homogeneous (at the higher temperature). Another observation related to isothermal crystallization of undercooled metallic melts and metallic glasses around the glass transition temperature [23], mentions that "...above Tg, the atomic transport is controlled by viscous flow rather than by diffusion as at temperatures below the glass transition .... thus leading to a very large increase in the nucleation rate above Tg". Moreover, according to Ref. [23],
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
metal-metalloid (Fe66Ni10B24) and transition metal (ZrsoCos0) glasses both crystallize by homogeneous nucleation above Tg, and by heterogeneous nucleation below Tg. Clearly, both nucleation modes have different activation energies. It can be inferred that if the heating rate in devitrification of a specific metallic system is high enough, the nucleation might be activated homogeneously. Conversely, heterogeneous nucleation will take place in that system when the heating rate is low. In that case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization, which itself has been induced by heterogeneous nucleation. If, as reported in the literature, the data analyses of many devitrification experiments [20] show serious deviations from the ' A r r h e n i u s ' behavior expected from the Kissinger/Avrami theory, crystallization has probably taken place by homogeneous nucleation, and probably at a large heating rate. An evident conclusion of this duality is the existence of a critical heating rate in non-isothermal crystallization processes. If the heating rate is below the critical rate, heterogeneous, site saturation, nucleation prevails. Conversely, if the heating rate is above that critical rate, the operative mode is homogeneous nucleation. The integral equation formalism [12-14] has shown that the KJMA rate equation is still valid in the case of heterogeneous nucleation taking place at randomly dispersed pre-existing nuclei, i.e., in the case referred to as 'site saturation'. It follows that the Kissinger multiple-scan technique and peak method [6,7], which conforms to the KJMA rate equation for assessing the kinetics in non-isothermal devitrification experiments [10], is applicable only to that specific case, very often encountered indeed in non-isothermal devitrification experiments. The limiting, critical heating rate can therefore be derived from the KJMA equation, which is the adequate formalism for heterogeneous (site saturation) nucleation. Let's define: ]b= (dT/dt). Eq. (8) may then be written as
~v(T,
t) = 1 - e x p [
4~ No[frG(T,)dT,] 3] 3 ~,3 [To J (9)
291
The temperature dependence of the growth rate is an Arrhenius type function of the activation energy, E c, for growth only of the crystalline phase:
10, where G o is a material parameter. Integration of Eq. (10) in the temperature interval of the non-isothermal devitrification process gives
frG(T)dT=Gofrexp[@-]dT, To
To
(11)
[
which can be approximated [24] as
- ] dT~- G°EG 10-2"315-0'4577(EG/RT). Gofrexp[ -EG ~r,,
[ Rr
R
(12) The relation between the heating rate, 7~, and the crystallized volume fraction, ~v(t), can now be formulated, after some algebraic manipulations i? = 7.7 × 10 - 3
GoEc -
-
10 -0"4577(EG/RT)
R
×
-No ln[1 - ~(t)] "
(13)
It can be shown by a simple computation that if the volume fraction transformed following heterogeneous nucleation, for which Eq. (13) is applicable, ranges between 99.999% and 99.900%, the contribution of the term ~ - 1/ln[1 - ~(t)] to the value of ik is 0.48 _ 0.04, meaning that the heating rate does not change by more than 10%. In the case of almost complete heterogeneous devitrification ( ~ v ( t ) > 99.999%), a critical heating rate, i/'crit,can be approximated as follows: Lrit = 4.4 X
lO-4GoEalO-°°55(ea/rc,03~o.
(14)
Eq. (14) shows that the critical heating rate depends strongly on both the values of the activation energy, E G, for growth of the crystalline phase and of the critical temperature. T~rit has to be chosen as the temperature for which the density of the homogeneously nucleated sites, as derived from classical thermodynamics, is significantly higher than the density of the pre-existing, heterogeneously nucleated
292
V. Erukhimovitch, J. Baram / Journal of Non-Crystalline Solids 208 (1996) 288-293
sites N0. In fact, Tcrit is the highest temperature at which the heterogeneous, site saturation, nucleation mode is still the dominant one. An indicator that the transition from the site-saturation heterogeneous nucleation mode to the thermally activated homogeneous nucleation mode indeed occurs, is when the Avrami exponent derived from the kinetic analysis of an isothermal experiment changes to a value higher than 3 (the value predicted by the crystallization kinetics theory for 3D, interface-controlled), as reported in Ref. [22], and others mentioned in the same paper 5. Again, if the heating rate is below the critical rate, Tcrit, heterogeneous site saturation nucleation prevails. Conversely, if the heating rate is above Tcrit' the operative mode in the considered transformation is mainly homogeneous nucleation. Experimental support for this finding may be deduced from the results reported in Ref. [21], fig. 6. Two extreme heating rates (by pulse heating and by ordinary DSC experiments, practiced on the same material system) resulted in entirely different traces (slopes) of the computed Kissinger plots. Calculated activation energies were reported to differ for rapid and slow heating rates: 290 kJ mo1-1 and 140 kJ mol -~, respectively. According to the model presented in this paper, the DSC data reflects heterogeneous nucleation, with site saturation. The involved activation energy in this case is the one for growth only of the crystalline phase. Conversely, in view of the above discussion, the dependence of peak temperatures on heating rates in the case of the pulse heating experiments, which probably reflect an homogeneous nucleation and growth process, cannot be analyzed by the Kissinger method, applicable only to heterogeneously nucleated transformations.
3. Conclusions In the framework of the Integral Equation formalism, an equation is derived for the kinetics of nonisothermal crystallization (devitrification). When the operative nucleation mode is of the heterogeneous,
5 For 3D diffusion-controlled transformations, the corresponding Avrami exponent is 3/2.
site saturation type, the new formalism reduces to the KJMA equation. It is shown that, as a result, the Kissinger multiple-scan technique used for the evaluation of activation energies, is not applicable to all non-isothermal transformations, but only to those where site saturation heterogeneous nucleation takes place. In a devitrification transformation, the heating rate controls the nucleation mode. If the heating rate is large enough, homogeneous nucleation is activated. Conversely, heterogeneous nucleation takes place when the heating rate is small. In that last case, the high temperatures imposed by the thermodynamics for homogeneous nucleation are not reached before completion of crystallization, which has been induced by heterogeneous nucleation. A critical heating rate has been derived, that depends on the amount of pre-existing nucleation sites, on the activation energy for growth of the new phase and on a critical temperature, the temperature at which the density of the homogeneously nucleated sites is significantly higher than the density of the pre-existing, heterogeneously nucleated sites.
Acknowledgements One of the authors (V.E.) acknowledges financial support by the Israeli Ministry of Science and Arts during the period of this research.
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