On the Classification of Phase Transformation ,SM,46,2002,893-898

8/12/2019 On the Classification of Phase Transformation ,SM,46,2002,893-898

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On the classication of phase transformationsJohn AAgren *

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 classication schemes, based on thermodynamics, microstructure or mechanism, are discussed andcriticized from a practical as well as a more fundamental point of view. For example, it is generally not meaningful toconsider rst and second-order transformations as equivalent with heterogeneous and homogeneous transformations,respectively. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Thermodynamics; Microstructure; Orderdisorder phenomena; Glass transition

1. Introduction

This report concerns a rather trivial discussionon the classication of phase transformations.Nevertheless, the discussion may be worthwhile toclarify some concepts and avoid unnecessary con-fusion in the theoretical analysis of phase trans-formations.

By tradition characterization and classicationof various phenomena are important ingredients inall elds of science. Such classication schemes areusually based on a physical picture of the phe-nomena under consideration and consequentlythey often have to be modied when new knowl-edge emerges. To be of any use, a classicationscheme must be sufficiently simple, i.e. it must bebased on a few clear concepts and easy to apply inpractice. When such a scheme is generally estab-lished, it is convenient to use and it may often yield

a deeper insight in various phenomena. For ex-

ample, the classication into diffusion controlledor diffusionless phase transformations is an excel-lent pedagogical approach when teaching harden-ing of steel on the undergraduate level. However,sometimes a certain classication may be lessmeaningful or even misleading and imply physicalrelationships that do not exist.

In the study of phase transformations threeclassication schemes are well established andtaught to students in materials science. They maybe called the thermodynamic, the microstructuraland the mechanistic classication scheme. We shallnow analyze each scheme from the conceptual aswell as the application point of view.

2. The thermodynamic classication scheme

This scheme was introduced by Ehrenfest [1] andis based on the behavior of the derivatives of theGibbs energy. Usually one considers a continuouschange in temperature under xed pressure. If the

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temperature change is slow enough to allow thesystem always to relax to internal equilibrium, theGibbs energy will be a continuous function of temperature. If the rst derivative o G =o T

P changes discontinuously at the transformationtemperature the transformation is of rst orderand, in general, if all n 1 derivatives are contin-uous and the nth derivative is discontinuous at thetransformation temperature, then the transforma-tion is of nth order. From basic thermodynamicswe may identify

o G =o T P S 1

o 2G =o T 2 P o H =o T P =T C P =T 2

where S , H and C P are the entropy, the enthalpyand the heat capacity, respectively. A rst-ordertransformation thus is accompanied by discontin-uous changes in entropy and enthalpy, i.e. exper-imentally there is a heat of transformation. Asecond-order transformation has a continuousvariation in entropy but a discontinuous change inheat capacity, i.e. a jump is observed in the heatcapacity. A well-known textbook example of arst-order transformation is the solidication of

pure Fe whereas its magnetic transformation at theCurie temperature is an example of a second-ordertransformation. The onset of long-range order inb-brass below the critical temperature is anotherexample of a second-order transformation. Whenplotting phase diagrams it is common to let dashedlines represent temperatures where there is a sec-ond-order transformation whereas the normalsolid lines represent rst-order transformations. Insome alloy systems ordering may be a rst-ordertransformation. For example, the AuCu phasediagram reveals regions of ordering at low tem-peratures. However, contrary to the case of b-brass the ordering is represented by solid lines andthere are two-phase elds representing a mixturebetween ordered and disordered material. In thesecomposition ranges the solid lines of the phasediagram indicates that ordering would be a rst-order transformation.

From a microstructural point of view solidi-cation occurs heterogeneously, i.e. by nucleationand growth of crystals, whereas ordering of the

second-order type occurs gradually and homoge-neously and there is no sharp interface betweenordered and disordered material. Thus it is com-mon to use the terms rst-order and heterogeneoustransformation as synonyms and second-order andhomogeneous transformation as synonyms.

The thermodynamic scheme is appealing due toits simplicity. The order of a transformation isrevealed, in principle, from calorimetric measure-ments provided that heating or cooling is slowenough. However, in practice the interpretation of an experimental heat effect is not always thatsimple. It is certainly impossible in most cases todistinguish between a second and third-order orhigher-order transformation and it is sometimeseven difficult to judge whether a transformation isof rst or second order. There is also a conceptualdifficulty since attention is paid to the particulartemperature where there is a discontinuity in thederivative. However, in the case of a typical sec-ond-order transformation, like ordering of b-brass, there is no change in structure at the criticaltemperature. Upon cooling there is a gradual in-crease in short-range order above and long-rangeorder below the critical temperature. The changein structure thus occurs gradually over a wide

temperature range rather than at a particulartransformation temperature.A more serious difficulty is that transformations

closely related to rst-order transformationsshould actually be classied as second-ordertransformations if the thermodynamic scheme isapplied strictly. For example, when a second ele-ment is added to a pure element, solidication oc-curs over a temperature range between the liquidusand solidus temperatures. In that range the molarentropy of the two-phase mixture is given by

S m f sS sm 1 f sS Lm 3

where S sm and S Lm are the molar entropies of the

solid and the liquid, respectively. Since the fractionof solid f s varies continuously with temperature, itis clear that also S m will be continuous. If we applythe thermodynamic classication scheme it is evi-dent that solidication of a binary alloy is not arst-order transformation. Taking the derivativeof the molar entropy we obtain

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o S mo T P f s

oS smo T P 1 f s

oS Lmo T P

S sm

S Lm

d f s

dT 4

The last term changes discontinuously at theliquidus and solidus temperatures and is zerooutside the solidication range. According to thethermodynamic scheme, the system thus has twosecond-order transformations, one at the liquidusand one at the solidus whereas the transformationactually occurs between the liquidus and solidus.In fact, all transformations in alloys occurring inconnection to a two-phase eld would be classiedas second-order transformations. Only the ones atinvariant temperatures, e.g. eutectic transforma-tions, would be of rst order. This problem withthe Ehrenfest scheme was discussed in some detailby Johnson and Voorhees [2].

This result leads to a number of surprisingconclusions. For example, the orderdisordertransformations in AuCu alloys, which are usu-ally regarded as rst-order transformations, areactually second-order transformations except atthe congruent transformation points at the stoi-chiometric compositions AuCu and AuCu 3. For

these compositions the two-phase elds collapse topoints and the transformation becomes of rstorder.

Obviously the above result is caused by theoccurrence of the two-phase eld in the phase di-agram. It is worth noticing that if we considersolidication of a pure element under constantvolume rather than constant pressure we wouldhave a solidication range rather than a particularsolidication temperature. Of course we shouldthen consider Helmholtz energy rather than Gibbsenergy and we would conclude that no transfor-mation could be of rst order.

On the other hand, one may ask if solidicationof a binary alloy AB would be of rst order if theexperiment is performed under constant chemicalpotential rather than composition. In the phasediagram with T and l B as axes the two-phase eldscollapse into lines and the fraction of solid wouldalways change discontinuously as such a line ispassed. Indeed, the transformation could thus beclassied as a rst-order transformation. However,

the classication should no longer be based on theGibbs energy but rather the function

U TS PV l B N B G l B N B l A N A 5

From the GibbsDuhem equation

S dT V d P N A d l A N B d l B 0 6

we have

o l Ao T P ;l B S = N A 7However, in general this would require a con-

tinuous change in the composition of the system.

In most cases such a condition would be difficult toarrange experimentally and therefore this possibleclassication is of little practical interest. It may bementioned that Hillert [3] has recently tried toovercome this difficulty by using a modied ver-sion of the Gibbs phase rule. He classies hetero-geneous phase transformations as sharp or gradualdepending on whether the phase eld, separatingthe two states, exists at a unique value or over arange of values of the variable used to accomplishthe transformation.

3. The microstructural classication scheme

The microstructural classication scheme isbased on the effect on the microstructure andstems from Gibbs [4]. In a heterogeneous phasetransformation the transformation occurs by themotion of a rather sharp interface between trans-formed material and material that has not yet beentransformed. Obviously, solidication of pure el-ements as well as alloys are classied in the sameway, i.e. as heterogeneous, when the microstruc-tural classication scheme is used. On the otherhand, a phase transformation is classied as ho-mogeneous 1 if it does not lead to any observable

1 Heterogeneous transformations occur by nucleation andgrowth. If the probability to form a nucleus is the sameeverywhere nucleation is said to be homogeneous. If nucleationis more likely to occur on heterogenities, e.g. interfaces andsurfaces, nucleation is said to be heterogeneous.

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heterogeneities, i.e. it occurs gradually to the sameextent everywhere. An important consequence of aheterogeneous phase transformation is that it muststart within some small volume and then spreadfrom there, i.e. it must proceed by nucleation andgrowth. The homogeneous transformation wouldoccur continuously without the need of any nu-cleation.

The microstructural classication scheme isappealing due to its conceptual simplicity. Anadvantage over the thermodynamic scheme is thatit seems to reect more of the physical character of a phase transformation. For example, the classi-cation is not changed if an alloy element is addedand it does not depend on whether a potential oran extensive variable is kept xed during the ex-periment. The ordering reaction in the AuCusystem would be classied as heterogeneous at allcompositions regardless of the change from rst tosecond order as the composition deviates from thestoichiometric ones. From a practical point of viewthe microstructural scheme has the disadvantagethat it requires a microstructural investigation of the material. There is also a conceptual problemthat may be illustrated by analyzing the transfor-mations inside a miscibility gap. As a material is

cooled inside the miscibility gap it will rst enter aregion of metastability where the decompositioninto the two phases will occur by nucleation andgrowth, i.e. the reaction is heterogeneous. How-ever, if the material is cooled further and no nu-cleation occurs it will enter into a region of instability and a reaction called spinodal decom-position will occur. This reaction was analyzed indetail by Hillert [5] adopting the nearest-neighborinteraction model. Hillerts thesis inspired Cahnand Hilliard [6] to undertake an elegant mathe-matical analysis of the spinodal decomposition. Amain result is that spinodal decomposition occursspontaneously without any nucleation and shouldthus be classied as homogeneous. However, it willeventually result in a heterogeneous structure andshould then be classied as heterogeneous. Or, inother words, the microstructural classicationscheme fails to catch the character of the spinodaldecomposition. On the other hand, all reactionsinside the miscibility gap would be classied assecond order regardless of their exact nature if the

thermodynamic classication scheme is applied. Itis worth noticing that we would have a similarsituation in the AuCu system if an alloy isquenched below the so-called ordering spinodal,which will appear somewhere in the ordered one-phase eld, and ordering would become homoge-neous.

4. The mechanistic classication scheme

The mechanistic classication scheme is basedon the detailed mechanism of a phase transfor-mation. Although it may rst seem very attractive,because it gives a physically based classication, ithas a number of drawbacks. It is conceptuallydifficult because there are so many mechanisms of phase transformations, see for example Table 1 inthe famous textbook by Christian [7]. Moreover, itis even more difficult to apply in practice becauseit does not only require detailed experimentalinvestigations but also cumbersome theoreti-cal considerations. For example, the distinctionbetween diffusion and interface-controlled trans-formations has been the source of endless con-

troversies. A reaction that may rst seem to beinterface controlled may require diffusion overshort distances and the rate may be determined bysome diffusion coefficient. It is then misleading toclassify it as diffusionless. The main problem withthis classication scheme is to analyze experimen-tal data on transformation rates and comparethem with theoretical estimates assuming diffusioncontrol or interface control. A precise distinctionwill depend on a precise knowledge about diffusioncoefficients, thermodynamic properties and theexact nature of the interfacial reactions. Thequestion is how to identify the process that con-sumes most of the available driving force. Acomplication may be that the transformation has amixed character. It must be emphasized that spe-cic crystallographic relationships over a phaseinterface do not necessarily indicate that its mi-gration is interface controlled. The interfacial re-actions may very well be so rapid that thetransformation rate is controlled, for example, bylong-range diffusion or heat transfer.



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5. Transformations that are not phase transforma-tions

The AlNi phase has a rather wide range of homogeneity and exhibits the same type of or-dering as b-brass. Like b-brass the orderingincreases with decreasing temperature but unlikeb-brass AlNi has some long-range order alreadywhen 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 ther-modynamic scheme would characterize the latterordering as a second-order transformation and theformer as no transformation at all. Only by hy-pothesizing a critical temperature above the melt-ing point, i.e. in metastable superheated AlNi, wewould characterize the ordering reaction of AlNias second order. The situation becomes even moreconfusing if we consider phases where the twosublattices are not equivalent. In that case therewill also be a gradual increase in order with de-creasing temperature but upon heating the long-range order will never disappear, not even if melting is neglected.

A similar phenomenon occurs when a liquid iscooled. If crystallization is avoided there is a

gradual loss of entropy, i.e. there is ordering. Theviscosity increases until, at some temperature T g,the liquid undergoes a so-called glass transition.The glass transition is accompanied, for example,by a drastic change in heat capacity and thermalexpansion. The glass transition thus has similarfeatures as a second-order transformation andshould be classied as such a transformation if thethermodynamic scheme is applied. This is proba-bly the main reason why the nature of the glasstransition has been the subject of numerous dis-cussions and misunderstandings. However, con-trary to other phase transformations, where thetransformation will be displaced to lower temper-atures the higher the cooling rate is, it is wellknown that T g is displaced to higher temperaturesby faster cooling. Consequently, many authorsconclude that the glass transition is not strictly asecond-order transformation. The general agree-ment rather seems to be that the glass transition isa kinetic phenomenon. However, Elliot [8] statesthere is also denitely a thermodynamic as-

pect . . . and Cohen and Grest [9] state Our the-ory thus leads to a rst-order phase transition orno transition at all.

The reason for this confusion must be that toomuch attention is paid to the glass transition tem-perature. As in the case of a homogeneous second-order transformation, nothing is really happeningat the temperature where the discontinuity in C P occurs. Long-range order develops below thetransformation temperature in b-brass and C P then jumps to quite high values. On the other hand,when a liquid is cooled from high temperaturesthere is a gradual loss in its entropy. As long as therelaxation processes are rapid enough to allow in-ternal equilibrium to be established, this processwill continue. The process should be denotedamorphous solidication because the amorphousliquid gradually transforms into an amorphoussolid. At some temperature relaxation becomesslow and the structure is frozen-in upon furthercooling, i.e. there is a drop in C P which marks theglass transition. The glass transition upon coolingthus marks the end of the real phase transforma-tion from liquid to amorphous solid rather than thetransformation itself. For metals this transforma-tion starts even above the normal melting point and

is often revealed by a gradual increase in heat ca-pacity upon cooling. It is worth emphasizing thatthe amorphous transformation is homogeneous if the microstructural classication scheme is applied.It is very difficult to apply the mechanistic classi-cation scheme because different liquids will havevery different relaxation processes on the molecularor atomic scale.

6. Conclusion

It is now time to completely abandon thecommonly used classication based on the orderof transformations. On the other hand, the dis-tinction between heterogeneous and homogeneoustransformations is clear and helpful and containsmuch of the physics although it is usually toocoarse to be of much practical use (transforma-tions in materials are usually heterogeneous). Aner division, e.g. the one by Christian [7], thus isneeded. However, in order to promote scientic

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progress and practical use such a ner classica-tion must be based on experimental facts, whichare reasonably easy to establish. The classicationshould also be based on the major practical con-sequences, e.g. is the transformation rapid or slow,does it involve change in composition, etc ratherthan assumed details of the reaction mechanism.

Acknowledgements

The author wishes to thank Dr. John Cahnfor many valuable discussions and much inspira-tion over the years. Professor Mats Hillert is ac-knowledged 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 metallicsolutions. 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.


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On the Classification of Phase Transformation ,SM,46,2002,893-898