Scripta Materialia 46 (2002) 893–898 www.actamat-journals.com
On the classification of phase transformations gren John A
*
Department of Materials Science and Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received 23 January 2002; accepted 12 March 2002
Abstract The various classification schemes, based on thermodynamics, microstructure or mechanism, are discussed and criticized from a practical as well as a more fundamental point of view. For example, it is generally not meaningful to consider first and second-order transformations as equivalent with heterogeneous and homogeneous transformations, respectively. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Thermodynamics; Microstructure; Order–disorder phenomena; Glass transition
1. Introduction This report concerns a rather trivial discussion on the classification of phase transformations. Nevertheless, the discussion may be worthwhile to clarify some concepts and avoid unnecessary confusion in the theoretical analysis of phase transformations. By tradition characterization and classification of various phenomena are important ingredients in all fields of science. Such classification schemes are usually based on a physical picture of the phenomena under consideration and consequently they often have to be modified when new knowledge emerges. To be of any use, a classification scheme must be sufficiently simple, i.e. it must be based on a few clear concepts and easy to apply in practice. When such a scheme is generally established, it is convenient to use and it may often yield
*
Tel.: +46-8-7909131; fax: +46-8-100411. gren). E-mail address:
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a deeper insight in various phenomena. For example, the classification into diffusion controlled or diffusionless phase transformations is an excellent pedagogical approach when teaching hardening of steel on the undergraduate level. However, sometimes a certain classification may be less meaningful or even misleading and imply physical relationships that do not exist. In the study of phase transformations three classification schemes are well established and taught to students in materials science. They may be called the thermodynamic, the microstructural and the mechanistic classification scheme. We shall now analyze each scheme from the conceptual as well as the application point of view.
2. The thermodynamic classification scheme This scheme was introduced by Ehrenfest [1] and is based on the behavior of the derivatives of the Gibbs energy. Usually one considers a continuous change in temperature under fixed pressure. If the
1359-6462/02/$ - see front matter Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 2 ) 0 0 0 8 3 - 0
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temperature change is slow enough to allow the system always to relax to internal equilibrium, the Gibbs energy will be a continuous function of temperature. If the first derivative ðoG=oT ÞP changes discontinuously at the transformation temperature the transformation is of first order and, in general, if all n 1 derivatives are continuous and the nth derivative is discontinuous at the transformation temperature, then the transformation is of nth order. From basic thermodynamics we may identify ðoG=oT ÞP ¼ S
ð1Þ
ðo2 G=oT 2 ÞP ¼ ðoH =oT ÞP =T ¼ CP =T
ð2Þ
where S, H and CP are the entropy, the enthalpy and the heat capacity, respectively. A first-order transformation thus is accompanied by discontinuous changes in entropy and enthalpy, i.e. experimentally there is a heat of transformation. A second-order transformation has a continuous variation in entropy but a discontinuous change in heat capacity, i.e. a jump is observed in the heat capacity. A well-known textbook example of a first-order transformation is the solidification of pure Fe whereas its magnetic transformation at the Curie temperature is an example of a second-order transformation. The onset of long-range order in b-brass below the critical temperature is another example of a second-order transformation. When plotting phase diagrams it is common to let dashed lines represent temperatures where there is a second-order transformation whereas the normal solid lines represent first-order transformations. In some alloy systems ordering may be a first-order transformation. For example, the Au–Cu phase diagram reveals regions of ordering at low temperatures. However, contrary to the case of bbrass the ordering is represented by solid lines and there are two-phase fields representing a mixture between ordered and disordered material. In these composition ranges the solid lines of the phase diagram indicates that ordering would be a firstorder transformation. From a microstructural point of view solidification occurs heterogeneously, i.e. by nucleation and growth of crystals, whereas ordering of the
second-order type occurs gradually and homogeneously and there is no sharp interface between ordered and disordered material. Thus it is common to use the terms first-order and heterogeneous transformation as synonyms and second-order and homogeneous transformation as synonyms. The thermodynamic scheme is appealing due to its simplicity. The order of a transformation is revealed, in principle, from calorimetric measurements provided that heating or cooling is slow enough. However, in practice the interpretation of an experimental heat effect is not always that simple. It is certainly impossible in most cases to distinguish between a second and third-order or higher-order transformation and it is sometimes even difficult to judge whether a transformation is of first or second order. There is also a conceptual difficulty since attention is paid to the particular temperature where there is a discontinuity in the derivative. However, in the case of a typical second-order transformation, like ordering of bbrass, there is no change in structure at the critical temperature. Upon cooling there is a gradual increase in short-range order above and long-range order below the critical temperature. The change in structure thus occurs gradually over a wide temperature range rather than at a particular transformation temperature. A more serious difficulty is that transformations closely related to first-order transformations should actually be classified as second-order transformations if the thermodynamic scheme is applied strictly. For example, when a second element is added to a pure element, solidification occurs over a temperature range between the liquidus and solidus temperatures. In that range the molar entropy of the two-phase mixture is given by Sm ¼ f s Sms þ ð1 f s ÞSmL
ð3Þ
where Sms and SmL are the molar entropies of the solid and the liquid, respectively. Since the fraction of solid f s varies continuously with temperature, it is clear that also Sm will be continuous. If we apply the thermodynamic classification scheme it is evident that solidification of a binary alloy is not a first-order transformation. Taking the derivative of the molar entropy we obtain
J. Agren / Scripta Materialia 46 (2002) 893–898
oSm oT
P
¼ fs
oSms oT
þ ð1 f s Þ P
df s þ ðSms SmL Þ dT
L
oSm oT
895
the classification should no longer be based on the Gibbs energy but rather the function
P
ð4Þ
The last term changes discontinuously at the liquidus and solidus temperatures and is zero outside the solidification range. According to the thermodynamic scheme, the system thus has two second-order transformations, one at the liquidus and one at the solidus whereas the transformation actually occurs between the liquidus and solidus. In fact, all transformations in alloys occurring in connection to a two-phase field would be classified as second-order transformations. Only the ones at invariant temperatures, e.g. eutectic transformations, would be of first order. This problem with the Ehrenfest scheme was discussed in some detail by Johnson and Voorhees [2]. This result leads to a number of surprising conclusions. For example, the order–disorder transformations in Au–Cu alloys, which are usually regarded as first-order transformations, are actually second-order transformations except at the congruent transformation points at the stoichiometric compositions AuCu and AuCu3 . For these compositions the two-phase fields collapse to points and the transformation becomes of first order. Obviously the above result is caused by the occurrence of the two-phase field in the phase diagram. It is worth noticing that if we consider solidification of a pure element under constant volume rather than constant pressure we would have a solidification range rather than a particular solidification temperature. Of course we should then consider Helmholtz energy rather than Gibbs energy and we would conclude that no transformation could be of first order. On the other hand, one may ask if solidification of a binary alloy A–B would be of first order if the experiment is performed under constant chemical potential rather than composition. In the phase diagram with T and lB as axes the two-phase fields collapse into lines and the fraction of solid would always change discontinuously as such a line is passed. Indeed, the transformation could thus be classified as a first-order transformation. However,
U TS þ PV lB NB ¼ G lB NB ¼ lA NA
ð5Þ
From the Gibbs–Duhem equation S dT V dP þ NA dlA þ NB dlB ¼ 0 we have olA ¼ S=NA oT P ;lB
ð6Þ
ð7Þ
However, in general this would require a continuous change in the composition of the system. In most cases such a condition would be difficult to arrange experimentally and therefore this possible classification is of little practical interest. It may be mentioned that Hillert [3] has recently tried to overcome this difficulty by using a modified version of the Gibbs phase rule. He classifies heterogeneous phase transformations as sharp or gradual depending on whether the phase field, separating the two states, exists at a unique value or over a range of values of the variable used to accomplish the transformation.
3. The microstructural classification scheme The microstructural classification scheme is based on the effect on the microstructure and stems from Gibbs [4]. In a heterogeneous phase transformation the transformation occurs by the motion of a rather sharp interface between transformed material and material that has not yet been transformed. Obviously, solidification of pure elements as well as alloys are classified in the same way, i.e. as heterogeneous, when the microstructural classification scheme is used. On the other hand, a phase transformation is classified as homogeneous 1 if it does not lead to any observable 1 Heterogeneous transformations occur by nucleation and growth. If the probability to form a nucleus is the same everywhere nucleation is said to be homogeneous. If nucleation is more likely to occur on heterogenities, e.g. interfaces and surfaces, nucleation is said to be heterogeneous.
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heterogeneities, i.e. it occurs gradually to the same extent everywhere. An important consequence of a heterogeneous phase transformation is that it must start within some small volume and then spread from there, i.e. it must proceed by nucleation and growth. The homogeneous transformation would occur continuously without the need of any nucleation. The microstructural classification scheme is appealing due to its conceptual simplicity. An advantage over the thermodynamic scheme is that it seems to reflect more of the physical character of a phase transformation. For example, the classification is not changed if an alloy element is added and it does not depend on whether a potential or an extensive variable is kept fixed during the experiment. The ordering reaction in the Au–Cu system would be classified as heterogeneous at all compositions regardless of the change from first to second order as the composition deviates from the stoichiometric ones. From a practical point of view the microstructural scheme has the disadvantage that it requires a microstructural investigation of the material. There is also a conceptual problem that may be illustrated by analyzing the transformations inside a miscibility gap. As a material is cooled inside the miscibility gap it will first enter a region of metastability where the decomposition into the two phases will occur by nucleation and growth, i.e. the reaction is heterogeneous. However, if the material is cooled further and no nucleation occurs it will enter into a region of instability and a reaction called spinodal decomposition will occur. This reaction was analyzed in detail by Hillert [5] adopting the nearest-neighbor interaction model. Hillert’s thesis inspired Cahn and Hilliard [6] to undertake an elegant mathematical analysis of the spinodal decomposition. A main result is that spinodal decomposition occurs spontaneously without any nucleation and should thus be classified as homogeneous. However, it will eventually result in a heterogeneous structure and should then be classified as heterogeneous. Or, in other words, the microstructural classification scheme fails to catch the character of the spinodal decomposition. On the other hand, all reactions inside the miscibility gap would be classified as second order regardless of their exact nature if the
thermodynamic classification scheme is applied. It is worth noticing that we would have a similar situation in the Au–Cu system if an alloy is quenched below the so-called ordering spinodal, which will appear somewhere in the ordered onephase field, and ordering would become homogeneous.
4. The mechanistic classification scheme The mechanistic classification scheme is based on the detailed mechanism of a phase transformation. Although it may first seem very attractive, because it gives a physically based classification, it has a number of drawbacks. It is conceptually difficult because there are so many mechanisms of phase transformations, see for example Table 1 in the famous textbook by Christian [7]. Moreover, it is even more difficult to apply in practice because it does not only require detailed experimental investigations but also cumbersome theoretical considerations. For example, the distinction between diffusion and interface-controlled transformations has been the source of endless controversies. A reaction that may first seem to be interface controlled may require diffusion over short distances and the rate may be determined by some diffusion coefficient. It is then misleading to classify it as diffusionless. The main problem with this classification scheme is to analyze experimental data on transformation rates and compare them with theoretical estimates assuming diffusion control or interface control. A precise distinction will depend on a precise knowledge about diffusion coefficients, thermodynamic properties and the exact nature of the interfacial reactions. The question is how to identify the process that consumes most of the available driving force. A complication may be that the transformation has a mixed character. It must be emphasized that specific crystallographic relationships over a phase interface do not necessarily indicate that its migration is interface controlled. The interfacial reactions may very well be so rapid that the transformation rate is controlled, for example, by long-range diffusion or heat transfer.
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5. Transformations that are not phase transformations The AlNi phase has a rather wide range of homogeneity and exhibits the same type of ordering as b-brass. Like b-brass the ordering increases with decreasing temperature but unlike b-brass AlNi has some long-range order already when it forms from the melt. Despite the fact that, upon cooling, we have exactly the same type of structural ordering in AlNi as in b-brass the thermodynamic scheme would characterize the latter ordering as a second-order transformation and the former as no transformation at all. Only by hypothesizing a critical temperature above the melting point, i.e. in metastable superheated AlNi, we would characterize the ordering reaction of AlNi as second order. The situation becomes even more confusing if we consider phases where the two sublattices are not equivalent. In that case there will also be a gradual increase in order with decreasing temperature but upon heating the longrange order will never disappear, not even if melting is neglected. A similar phenomenon occurs when a liquid is cooled. If crystallization is avoided there is a gradual loss of entropy, i.e. there is ordering. The viscosity increases until, at some temperature Tg , the liquid undergoes a so-called glass transition. The glass transition is accompanied, for example, by a drastic change in heat capacity and thermal expansion. The glass transition thus has similar features as a second-order transformation and should be classified as such a transformation if the thermodynamic scheme is applied. This is probably the main reason why the nature of the glass transition has been the subject of numerous discussions and misunderstandings. However, contrary to other phase transformations, where the transformation will be displaced to lower temperatures the higher the cooling rate is, it is well known that Tg is displaced to higher temperatures by faster cooling. Consequently, many authors conclude that the glass transition is not strictly a second-order transformation. The general agreement rather seems to be that the glass transition is a kinetic phenomenon. However, Elliot [8] states ‘‘there is also definitely a thermodynamic as-
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pect. . .’’ and Cohen and Grest [9] state ‘‘Our theory thus leads to a first-order phase transition or no transition at all’’. The reason for this confusion must be that too much attention is paid to the glass transition temperature. As in the case of a homogeneous secondorder transformation, nothing is really happening at the temperature where the discontinuity in CP occurs. Long-range order develops below the transformation temperature in b-brass and CP then jumps to quite high values. On the other hand, when a liquid is cooled from high temperatures there is a gradual loss in its entropy. As long as the relaxation processes are rapid enough to allow internal equilibrium to be established, this process will continue. The process should be denoted amorphous solidification because the amorphous liquid gradually transforms into an amorphous solid. At some temperature relaxation becomes slow and the structure is frozen-in upon further cooling, i.e. there is a drop in CP which marks the glass transition. The glass transition upon cooling thus marks the end of the real phase transformation from liquid to amorphous solid rather than the transformation itself. For metals this transformation starts even above the normal melting point and is often revealed by a gradual increase in heat capacity upon cooling. It is worth emphasizing that the amorphous transformation is homogeneous if the microstructural classification scheme is applied. It is very difficult to apply the mechanistic classification scheme because different liquids will have very different relaxation processes on the molecular or atomic scale.
6. Conclusion It is now time to completely abandon the commonly used classification based on the order of transformations. On the other hand, the distinction between heterogeneous and homogeneous transformations is clear and helpful and contains much of the physics although it is usually too coarse to be of much practical use (transformations in materials are usually heterogeneous). A finer division, e.g. the one by Christian [7], thus is needed. However, in order to promote scientific
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progress and practical use such a finer classification must be based on experimental facts, which are reasonably easy to establish. The classification should also be based on the major practical consequences, e.g. is the transformation rapid or slow, does it involve change in composition, etc rather than assumed details of the reaction mechanism.
Acknowledgements The author wishes to thank Dr. John Cahn for many valuable discussions and much inspiration over the years. Professor Mats Hillert is acknowledged for many valuable suggestions.
References [1] Ehrenfest P. Leiden Commun Suppl 1933;75b. [2] Johnson WC, Voorhees PW. Acta Metall Mater 1990;38:1183. [3] Hillert M. Phase equilibria, phase diagrams and phase transformations––their thermodynamic basis. Cambridge: Cambridge University Press; 1998. [4] Gibbs JW. The collected works. Thermodynamics, vol. I. Yale University Press, 1948. [5] Hillert M. A theory of nucleation for solid metallic solutions. DSc thesis, MIT, 1956. [6] Cahn JW, Hilliard JE. J Chem Phys 1958;28:258. [7] Christian JW. The theory of transformations in metals and alloys. Oxford: Pergamon Press; 1965. [8] Elliott SR. Physics of amorphous materials. London: Longman House; 1983. [9] Cohen MH, Grest GS. Phys Rev B 1979;20:1077.
