Classification of Phase Transformations

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:89

Trim:165 mm×240 mm

Integra, India

Chapter 2

Classification of Phase Transformations Symbols G: T: Tc : S: P: H: Cp : : Tm : -phase: -phase: ∗ : e : A : Cij : Ms : V: Vi , Gid : Vd , Gdd :

2.1

and Abbreviations Gibbs free energy ( G = H-TS) Temperature Equilibrium transition temperature Entropy Pressure Enthalpy Specific heat at constant pressure Generalized order parameter Melting temperature hcp phase in Ti and Zr based alloys bcc phase in Ti and Zr based alloys Order parameter corresponding to maximum in free energy Equilibrium order parameter for a first order transition Chemical potential of component, A in the phase,  Elastic stiffness modulus (elastic constant) Temperature at which martensite starts forming during quenching Specific volume Velocity and dissipated free energy associated with interface process Velocity and dissipated free energy associated with diffusion process

INTRODUCTION

The study of phase transformations is of interest to metallurgists, geologists, chemists, physicists and indeed to all scientists concerned with the states of aggregation of atoms. Due to the multidisciplinary interest in this subject, a wide variety of nomenclature, sometimes even misleading, has been introduced in the literature for the characterization of different types of phase transformations. It is not uncommon that different sets of terminologies are used in different disciplines for describing essentially similar phase transformations which, in a generalized manner, can be defined as a change in the macrostate of an assembly of interacting atoms or molecules as a result of some variation in the external constraints. The diversity of scientific interest and the complexity of the possible interactions between individual atoms of the assembly naturally lead to many different 89

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

90

5-6-2007

9:27 a.m.

Page:90

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

approaches for the study of phase transformations. Physicists primarily focus their attention on higher-order continuous phase transitions in single-component systems such as magnetic, superconducting and superfluid transitions. In contrast, metallurgists and chemists are mainly concerned with phase transformations (which include phase reactions) involving changes in crystal structure, chemical composition and order parameter (both long and shortrange). Phase transformations encountered by geologists, though quite similar to those observed in metallic and ceramic systems, usually occur over much more extended time and length scales under extreme conditions of pressure and temperature. Phase transformations also occur in organic materials such as polymers, biological systems and liquid crystals. Many of the relevant concepts developed for inorganic systems have parallels in organic systems. However, no attempt will be made in this chapter to compare and contrast phase transformations in organic and inorganic systems as the nature of atomic interactions responsible for the transformations is quite different in these two classes of materials. Alloys, intermetallics and ceramics form a group of materials in which phase transformations can be discussed on a common conceptual basis and, therefore, a single classification scheme can be used for appropriately grouping different types of transformations in these systems. As mentioned earlier, Ti- and Zr-based systems, which include alloys, intermetallics and ceramics, exhibit nearly all possible types of phase transformations and, therefore, serve as excellent examples for studies on phase transformations in inorganic materials in general. Phase transformations can be classified on the basis of different criteria, namely, thermodynamic, kinetic and mechanistic (Christian 1965, Roy 1973, Rao and Rao 1978). A comparison of the characteristic features of different types of transformations is presented in this chapter with a view to providing a coarse-brush picture of these in a generalized manner. The chapters which follow will describe these transformations more elaborately, taking illustrative examples from Ti- and Zr-based systems.

2.2

BASIC DEFINITIONS

In order to resolve some of the confusion and controversy which are of a semantic nature a summary of some basic definitions is presented here. A phase is a portion of a system bounded by surfaces and with a distinctive and reproducible structure and composition. Within a single phase, minor fluctuations in structure and/or in composition can occur. One phase can be distinguished from a second phase if at the contacting surface there is a sharp (within one or two atom layers) first-order change in composition and/or structure and hence properties.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:91

Classification of Phase Transformations

Integra, India

91

The two terms phase transformation and phase transition are often used interchangeably. Sometimes a distinction between them is implied but rarely specified. In a paper entitled “A synchretist classification of phase transitions", Rustum Roy (1973) has addressed the controversy which exists in the literature in this regard. Generally the word phase transition is restricted to transitions between two phases which have identical chemical compositions, while the term phase transformation covers a wider spectrum of phenomena which include phase reactions leading to compositional changes. In the metallurgical literature, phase transformations include precipitation of a second phase, , of a different crystal structure and chemical composition from the parent  phase ( →  + ),  and  having the same crystal structure but different chemical compositions, eutectoid decomposition ( →  + ) and many such processes which, in the chemistry literature, are grouped as phase reactions (Rao and Rao 1978). A more subtle point concerns the meaning of identical chemical composition. The equilibrium point defect concentration may be different in two polymorphs. Though in a strict sense they cannot be considered as identical in chemical composition, transformations between such polymorphs are usually classified as composition-invariant transformations. In considering equilibria between two phases, the requirement of reversibility must be taken into account. Several relationships pertaining to equilibria between two phases can be explained using the free energy versus temperature plot of a single component system (Figure 2.1). The liquid (L) to crystal (A) transition, Equ

ilibr Gla ium ss

A′

tas

tab

oo

le A

rc

TA/B

led

B Cry

stal

uid

liq

Crystal

pe

Me

Su

Free energy

Tg

A

T L/A

Enantiotropic

Liq

Monotropic

T2

uid

T1

Temperature

Figure 2.1. Free energy versus temperature plots showing phase transformations in a singlecomponent system. The differences between monotropic and enantiotropic transitions and between stable equilibrium and metastable equilibrium transitions are highlighted.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

92

5-6-2007

9:27 a.m.

Page:92

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

L  A, at the melting point, TL/A , and the crystal A to crystal B transition, A  B, at the transition temperature, TA/B , are stable equilibrium transitions. The transition between a metastable phase A and another metastable phase A , A (metastable)  A (metastable), can also be an equilibrium transition and can be grouped along with the former two cases as being enantiotropic, i.e. reversible and governed by classical thermodynamics. In contrast, when we consider transitions which can be represented by vertical lines in this diagram, such as A (metastable) → B (stable) at T1 and Glass → A (stable) at T2 , the reversibility criterion is not met. These irreversible transformations, defined as monotropic transitions, proceed only in one direction and it is not possible to establish an equilibrium between the parent and the product phases. Polymorphic transformations are generally defined as those which involve a structural transition without a change in the chemical composition. Sometimes these transformations are also referred to as congruent processes. There are, however, several examples, such as the transformation of crystalline oxygen to crystalline ozone and transformations of position isomers, which satisfy the aforementioned definition of a polymorphic transition, but cannot even be considered as phase transitions. This is because ozone and oxygen, in the phase rule sense, are two different substances (or components) which survived even the solid → liquid → vapour transitions while preserving their individuality. Similarly each position isomer is an individual component and, therefore, isomeric transitions cannot be considered as phase transitions. In view of this, the definition of polymorphic transformations needs to be restricted to transformations involving phases with different crystal structures which are part of a single component system. In multicomponent metallic alloys and intermetallics, chemical composition-invariant crystallization is a good example in which the parent phase transforms to the product without allowing any partitioning of the constituent elements (or components) between the two phases. In this sense, the system behaves as if it is a single-component system.

2.3

CLASSIFICATION SCHEMES

There are several ways in which phase transformations can be classified, based on thermodynamic, kinetic and mechanistic criteria. A single classification scheme may not be adequate to include all types of transformations encountered in all varieties of materials. In this chapter, an attempt is made to evolve a classification scheme which is applicable to phase transformations in metals, alloys, intermetallics and ceramics.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:93

Classification of Phase Transformations

Integra, India

93

2.3.1 Classification based on thermodynamics Ehrenfest (1933) proposed a classification based on the successive differentiation of a thermodynamic potential, usually the Gibbs free energy function, with respect to an external variable such as temperature or pressure. The order of a transformation is then given by the lowest derivative which shows a discontinuity at the transition point. In the generalized sense, for an nth order transition 

n G

T n





n−1 G

T n−1

= 0 P

 =0

at

T = Tc

(2.1)

where G represents Gibbs free energy and Tc is the equilibrium transition temperature. It is to be noted that the Ehrenfest classification can be used only for equilibrium transitions of a single component system. Substituting n = 1 and 2 in Eq. (2.1), at T = Tc we get for first-order transitions 

G = 0

G

T

and for second-order transitions    

G H = − S = − = 0

T P Tc

 = − S = P



2 G

T 2

 P

H Tc

(2.2)

  Cp 1

H = = = 0 Tc

T P Tc (2.3)

A comparison between a first- and a second-order transition can be made in schematic plots of different thermodynamic quantities as functions of temperature (Figure 2.2). First-order transitions are characterized by discontinuous changes in entropy, enthalpy and specific volume. The change in enthalpy corresponds to the evolution of a latent heat of transformation, and the specific heat at the transition temperature, as a consequence, is effectively infinite. In contrast, second-order transitions are characterized by the absence of a latent heat of transformation (as H, S and V do not undergo a discontinuous change at Tc ) and a high specific heat at the transition temperature. There are experimental results which show that in some instances of second-order transitions the specific heat at Tc exhibits infinity rather than a finite discontinuity. A true second-order transition is, therefore, defined as one showing a finite discontinuity in the second derivative of the Gibbs function while a so-called lambda point transition exhibits an infinity. Though the Ehrenfest classification examines the presence of a discontinuity in the nth derivative of the Gibbs function for deciding the order of a transition, in the modern literature transitions with n ≥ 2 are grouped

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

94

5-6-2007

9:27 a.m.

Page:94

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys First order

G

Second order

S

G L TM

T

Tc

T

Tc

T

Tc

T

L H

q

S TM

H

T CP

CP

TM

T Temperature

Figure 2.2. Changes in the thermodynamic quantities, free energy, enthalpy and specific heat, at the transition temperatures corresponding to first-order and second-order phase changes.

together as “higher-order transitions” which are characterized by a continuous first derivative, G/ T P = 0, followed by either a discontinuity or infinity for “higher” derivatives. In a multidimensional plot of free energy against temperature, pressure, etc. each phase can be represented by a well-defined surface, as illustrated in Figure 2.3. The equilibrium transformation conditions between two phases are then defined by the intersection of two such surfaces. Moreover the free energy surface for a given phase may be extrapolated into conditions where that phase is not in thermodynamic equilibrium, and the difference in free energy, which is represented by the separation of the free energy surfaces corresponding to the two phases, can be regarded as the driving force for a first-order transformation from one phase to the other. This concept of a metastable phase is not readily applicable to a second-order transformation where it is more appropriate to consider that there is a single continuous free energy surface. Most of the phase transformations encountered in metallic systems are of the first-order type. Ferromagnetic ordering and some chemical ordering processes are examples of higher-order transitions in metallic systems. These transitions can be represented in “mean field” descriptions of cooperative phenomena where the respective order parameters continuously decrease to zero as the temperature

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:95

Classification of Phase Transformations

Integra, India

95

β

Free energy

α

e

ur

ss

e Pr

α β Temperature

Figure 2.3. Free energy surfaces for two phases,  and , as functions of pressure (P) and temperature (T ). The projection of the line of intersection of the two surfaces on the P–T plane represents the – phase boundary in the P–T phase diagram.

is raised to the transition temperature (Curie temperature or the critical ordering temperature), as shown in Figure 2.4. Any transition which can be described in terms of a continuous change in one or more order parameters can be treated in terms of a generalized Landau equation (Landau and Lifshitz 1969) which states

Tc

η

η Tc

T

T First order

Second order (a)

(b)

Figure 2.4. Order parameter () versus temperature (T ) plots for (a) second-order and (b) first-order transitions.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

96

5-6-2007

9:27 a.m.

Page:96

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

that close to the critical temperature the free energy difference, G, between finite and zero values of the order parameter, , may be expanded as a power series: G = A2 + B3 + C4 + · · ·

(2.4)

the coefficients A, B, C, etc. being functions of pressure and temperature. The fundamental differences between the first- and higher-order transitions can be explained on the basis of the corresponding Landau plots. For higher-order transitions, the free energy must be an even function of  which means that B = 0 (and similarly the coefficients of the odd-powered terms of  are zero). Figure 2.5(a) shows the G versus  plots for higher-order transitions at different temperatures, both above and below Tc . When T > Tc , the system exhibits a single stable equilibrium at  = 0 which corresponds to a positive value of A. As the temperature approaches Tc , the curvature ( 2 G/ 2 at  = 0 gradually decreases and as the temperature is lowered below Tc , the curvature as well as the value of A become negative. This essentially means that the system becomes unstable at T = Tc and any infinitesimal fluctuation in the order parameter leads to a lowering First order

Second order

T ≈ Tc Tc > T

T > Tc Free energy

Free energy

T > Tc

η

0

T ≈ Tc Tc > T > Ti

η

0

η = ηc Ti > T

–ve +ve Tc >> Ti

Tc = Ti

(b)

(a)

Figure 2.5. Free energy as a function of order parameter () for (a) second-order and (b) firstorder transitions. In the case of second-order transitions, the parent phase becomes unstable, ( 2 G/ 2 ) < 0 at  = 0 at the transition temperature, Tc , which is the same as the instability temperature, Ti . For some first-order transitions an instability temperature, Ti (which is much lower than the equilibrium transition temperature, Tc ), can be identified where the parent phase becomes unstable at  = 0.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:97

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

97

of the free energy. The negative curvature of the G versus  plot also implies that with an increase in the order parameter, the free energy progressively decreases, finally reaching the stable equilibrium positions defined by the minima present in the plots corresponding to T < Tc . The two minima corresponding to the positive and negative  values represent two equivalent states associated with antiphase domains of the same ordered structure. Landau plots for a first-order transition are shown in Figure 2.5(b). In this case, the value of B in the Landau expansion (Eq. (2.4)) is not zero. At the transformation temperature, the G versus  curve shows two minima, one at  = 0 and the other at  = c , the two minima being separated by a free energy barrier. At  = 0 the system is not unstable as the curvature remains positive at this point at T = Tc . A continuous increase in the order parameter, therefore, will initially raise the free energy which will drop only after the peak of the free energy hill is crossed. Since the system as a whole is not unstable either at Tc or at temperatures close to but below Tc , a gradual transition of the system in a homogeneous manner to the free energy minimum at or near  = c is not possible. A phase transition under such a situation can initiate only if localized portions of the system are activated to cross the free energy barrier to reach a point beyond ∗ where  can grow further spontaneously. The formation of such localized product phase regions (where  has nearly reached the c value) is known as nucleation. The product nuclei remain separated from the parent phase by sharp interfaces and the phase transition proceeds through the growth of these nuclei. The presence of two free energy minima separated by a free energy hill near the equilibrium in the case of a first-order transformation brings out its characteristic features, namely, the coexistence of the parent and the product phases and the discrete nature (involving nucleation and growth) of the transformation. In contrast, all higher-order transitions, by definition, are homogeneous in the sense that the parent and the product phases cannot be distinguished at any stage of the transition and there is no question of having an interface between the two phases. It is interesting to note that a discussion on Landau’s theory, which is strictly concerned only with the equilibrium state, has led us to consider continuous vis-à-vis discrete transformations. Continuous or homogeneous transitions are those in which the parent phase as a whole gradually evolves into the product phase without creating a localized sharp change in the thermodynamic properties and the structure in any part of the system. Such a process can occur only when the system becomes unstable with respect to an infinitesimal fluctuation which leads to the transition and the free energy of the system continuously decreases with the amplification of such a fluctuation. All higher-order transitions, by definition, satisfy the condition of homogeneous/continuous transformation at equilibrium. In contrast, all first-order

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

98

5-6-2007

9:27 a.m.

Page:98

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

transitions at equilibrium are discrete transitions which necessarily involve nucleation and growth. When we consider Landau plots for first-order transitions at temperatures far below Tc (Figure 2.5(b)) we notice that at temperatures below the instability temperature, Ti , the curvature of the G versus  plot becomes negative at  = 0 and the free energy continuously drops with increase in , finally reaching the value corresponding to c . Therefore below Ti a first-order transition can also proceed in a continuous mode. It must be emphasized here that the occurrence of an instability temperature is not universal for all first-order transitions. Only in rather a limited number of cases can first-order transitions be described in terms of a Landau representation. Spinodal clustering, spinodal ordering and displacement ordering processes are some examples in which continuous first-order transitions are encountered in conditions far from equilibrium. Equilibrium phase diagrams showing a miscibility gap correspond to solid solutions which exhibit a clustering tendency. The boundary of the equilibrium two-phase field, 1 + 2 , in the phase diagram (Figure 2.6) is determined by equating the chemical potentials  of the two components, A and B, in the two phases, 1 and 2 , in equilibrium at a given temperature, T1 : 1 2 = AB AB





(2.5)

α A

Temperature

α1

X1

X 2 α2

B

T1

Equilibrium solvus Coherent solvus Coherent spinodal

C D

Atom fraction (X)

Figure 2.6. Schematic phase diagram for a clustering system which remains a homogeneous solid solution in region A. A phase separation process occurs by a discrete nucleation and growth mechanism in regions B and C. Spinodal decomposition occurs in region D.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:99

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

99

The concentrations, X1 and X2 , at which the tie line at T1 intersects the miscibility gap correspond to those of 1 and 2 . A homogeneous solid solution  decomposes into an incoherent mixture of 1 and 2 as it is brought from the region A to the region B. Such a decomposition reaction, involving the creation of sharp interfaces between the two phases, is undoubtedly of the first order. A coherent mixture of the two phases can exist in the region C defined by the coherent miscibility gap. The coherent spinodal region D, fully residing within the region C, defines the concentration–temperature field where the phase separation process can initiate by introducing long wavelength (relative to interatomic distances) concentration fluctuations and a continuous amplification of such fluctuations. Though the process is continuous, the transformation is of the first order except at the point where the coherent spinodal touches the coherent solvus, where it may be considered to be of second order. The driving force for the phase separation arising from the negative curvature of the free energy (G)–concentration (X) plot (( 2 G/ X 2 )< 0 in the spinodal region) is opposed by the gradient energy and the coherency strain energy. All these factors and a correction factor for thermal fluctuations determine the wave number for which the amplification rate is the highest. Conceptually a chemical ordering process in which the ordered superlattice can be created only by replacement of atoms in the lattice of the disordered phase can be described in a manner similar to the spinodal clustering process. In the case of continuous ordering, concentration modulations with wavelengths of the order of the interatomic spacing need to be introduced. For an ordering system, the effective gradient energy is negative, and many of the predictions are opposite to those of spinodal decomposition. The amplification factor is negative beyond a critical wavelength and the maximum amplification corresponds to a wavelength equal to a small lattice vector. As shown in Figure 2.7(a) and (b), continuous ordering can be envisaged for both first-order and second-order reactions. Any change in the lattice dimensions due to ordering introduces a third-order term in the Landau equation (Eq. (2.4)) which makes the transformation first order. Continuous ordering in the first-order case requires finite supercooling below the coherent phase boundary. De Fontaine (1975) has also distinguished spinodal ordering from continuous ordering. In the former, the early stages of ordering are characterized by the ordering wave vectors which maximize the amplification factor but the amplification of these does not evolve the equilibrium ordered structure. In true continuous ordering, the equilibrium ordered structure continuously evolves from a low-amplitude quasihomogeneous concentration wave. De Fontaine (1975) has also examined the Landau–Lifshitz symmetry rules for determining whether a specific ordering wave vector qualifies to be a candidate for a second-order transformation.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

100

5-6-2007

9:27 a.m.

Page:100

Integra, India

Phase Transformations: Titanium and Zirconium Alloys First order

Second order αd

Temperature

Trim:165 mm×240 mm

αd Equilibrium phase boundary

A

A

Coherent phase boundary B

B C

αo

Coherent instability boundary, T i

D

X

X (a)

(b)

Figure 2.7. Schematic phase diagram for an ordering transition of (a) second order and (b) first order in which continuous ordering occurs under supercooling below the instability temperature, Ti . Region A represents the phase field where the disordered solid solution, d , is stable while Region B corresponds to the stability domain of two-phase mixture, o + d . Region C corresponds to stability domain of ordered o . In the case of a second-order process the d → o transition occurs in a continuous manner below the equilibrium phase boundary. In contrast, continuous ordering is possible in Region D only below the coherent instability boundary, Ti , in some first-order transitions (where the symmetry elements of the ordered structure form a subset of those of the parent disordered structure).

A product phase can evolve from the parent phase through a continuous displacement of the parent lattice. Two types of displacement, namely, a homogeneous lattice deformation and a relative displacement of atoms within unit cells (often called shuffles), can take place, either singly or in combination. The lattice deformation produces a change in volume and external shape and it seems very improbable that this could be accomplished continuously or homogeneously unless the principal lattice strains are very small (within the limits of linear elastic strain). Most martensitic transformations in metals and alloys involve much larger values of lattice strains and are, therefore, not candidates for continuous transformations. However, the possibility of a continuous displacive transformation has to be considered if the lattice deformation is very small. Transformations in some ferroelectric crystals of low symmetry are believed to be of the second order and occur by the progressive development of an instability in a dynamic plane displacement which becomes, at Tc , a static wave extending through the crystal. The approaching soft-mode instability is indicated as a pretransition effect above the transition temperature by a reduction in an appropriate lattice stiffness. Low and reducing values of the shear constant, 21 (C11 − C12 ), are

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:101

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

101

found in the parent phase as Ms is approached from higher temperatures in many noble metal bcc alloys and in fcc In–Tl; but in other martensitic transformations, especially those in steels, there are no anomalies of this kind and, therefore, these transformations are clearly of the first order. The athermal bcc →  transformation, which is envisaged as a pure shuffle transition, is another example of a continuous displacive transformation which is a first-order transformation from the consideration of symmetry rules. Though it is possible to accomplish this transformation by a continuous amplification of a displacement wave, the observed fine particle and dual phase ( + ) morphology suggests that -particles form in a quasiperiodic manner through heterophase fluctuations which have the form of ellipsoidal wave packets of displacement wave (Cook 1974). 2.3.2 Classifications based on mechanisms Buerger (1951) has introduced a classification based on mechanisms, namely, reconstructive and displacive transformations. In the metallurgical literature, however, a mechanistic classification groups transformations into (a) nucleation and growth and (b) martensitic types. The introduction of the term nucleation and growth in this context has created considerable confusion, as all first-order transformations including martensitic transformations require the nucleation and the growth steps. In the current literature, usage of such confusing nomenclature is avoided and a mechanistic classification designates the two classes as (a) diffusional and (b) displacive transformations. The former corresponds to the reconstructive transformation in which atom movements from the parent to the product lattice sites occur by random diffusional jumps. This implies that near neighbour bonds are broken at the transformation front and the product structure is reconstructed by placing the incoming atoms at appropriate positions which results in the growth of the product lattice. In contrast, atom movements in a displacive transformation can be accomplished by a homogeneous distortion, shuffling of lattice planes, static displacement waves or a combination of these. All these displacive modes involve cooperative movements of large numbers of atoms in a diffusionless process. Displacive (which includes martensitic) transformations initiate by the formation of nuclei of the product phase, and the growth of these nuclei occurs by the movement of a shear front at a speed that approaches the speed of sound in the material under consideration. In order to differentiate the mechanisms of atom movements across the transformation front, Christian (1965, 1979) has compared the movements involved in diffusional and displacive transformations with civilian and military movements, respectively. In the latter case, if the atoms are labelled in the parent lattice, the coordination between the neighbours can be shown to be essentially retained in

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

102

5-6-2007

9:27 a.m.

Page:102

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys Shear F

F

E

E

D

D

C

C

B

B A

G

H

I

A

J

G

H

I

J

(a) Shear

N′

N M

P′

O′

P O

M′ Q

R

Q′

S

(b)

Figure 2.8. Schematic illustration of lattice correspondence in a two-dimensional lattice. Though the parent and product lattices in both (a) and (b) are identical, the lattice correspondences in the two cases can be distinguished if the dots representing atoms occupying the lattice sites can be labelled. It is through the establishment of the lattice correspondence that the nature of the homogeneous shear and shuffle, if required, can be identified.

the product lattice, though the bond angles undergo changes. This point is illustrated in Figure 2.8(a) which shows how a set of atoms (labelled A, B, C, etc.) decorating the parent lattice changes to the product lattice. The existence of a lattice correspondence implies that a vector in the parent lattice, defined by the sequence of atoms ABCD   , becomes a vector in the product lattice with the atoms arranged in the same sequence, although the spacing between them gets altered to match the product lattice dimensions. Such a transformation can be viewed as a homogeneous deformation of the parent lattice (a simple shear in the

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:103

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

103

case of the two-dimensional illustration in Figure 2.8(a), shear directions being shown by arrows). The importance of lattice correspondence in determining the shear can be illustrated by Figure 2.8(b) which shows identical arrays of spots representing the parent and the product structures but with a different lattice correspondence. The atomic rows MNO and MQR are shown in both the parent and the product structures. It is evident from this drawing that the product structure is derived by a combination of a homogeneous deformation and a shuffle. The shear, as indicated by arrows, transforms the rectangle MOPQ in the parent to a parallelogram M O P Q in the product and the atom N is shifted to its new position by a shuffle. The best experimental evidence for the inheritance of the atomic coordination through a displacive transformation is provided by the observation that the chemical order present in the parent structure is fully retained in the product structure. A similar correspondence also exists for crystallographic planes. A relationship of this kind in which straight lines transform to straight lines and planes to planes is described mathematically as an affine transformation. Physically it may be considered as a homogeneous deformation of one lattice into the other. The correspondence associates each vector, plane and unit cell of the parent with a corresponding vector, plane and unit cell of the product. In general, the corresponding lattice vectors and the spacings of corresponding lattice planes are not equal in the two structures, and the angular relation between any pair of lattice vectors in the parent structure is not preserved in the product. It is to be noted that the lattice correspondence does not by itself imply any orientation relation between the phases, since the transformation may involve a rigid body rotation of the product structure with respect to the parent structure. In diffusional transformations, such lattice correspondences are not present. Even in those cases of diffusional transformations in which the chemical compositions of the parent and the daughter phases are identical and a strict orientation relationship exists between them, random jumps of atoms from the parent to the product lattice positions do not permit the lattice correspondence to be preserved. Such composition-invariant diffusional transformations proceed by atomic jumps across the advancing transformation fronts which separate the parent and the product phases. In order to understand the basic difference between displacive and diffusional transformations let us again consider labelling the atoms as A, B, C, D, etc. in the parent lattice and the same set of atoms as A , B , C , D etc. in the product lattice in Figure 2.9(a) and (b), respectively. The transformation front has been shown to advance by a single atomic layer. This schematic drawing shows that the sequence in which these atoms were placed before the transformation is not the same as that in the product lattice. This can happen if each atom breaks the bonds with its neighbours in the parent lattice and shifts to a new position

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

104

5-6-2007

9:27 a.m.

Page:104

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

F′

F

E′

E

D′

D C B

C′ B′

A (a) Transformation front F′

F

D′

E

E′

D

C′

C

A′

B

B′ A (b)

Figure 2.9. Schematic diagram of atom movements across the transformation front in a (a) displacive transformation; (b) diffusional transformation.

which corresponds to a lattice point in the product structure. In this manner, the transformation boundary proceeds towards the parent phase, converting the parent to the product phase. The jumps of atoms A, B, C, etc. are random and are not correlated with those of their neighbours, unlike the case of displacive transformations. Since diffusional transformations involve the breaking of bonds between neighbouring atoms and the reconstruction of bonds to form the product phase structure, they are also known as reconstructive transformations.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:105

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

105

2.3.3 Classification based on kinetics Phase transformations are also grouped in terms of the kinetics of the process. The most important distinguishing kinetic criterion is the requirement of thermal activation. First-order transformations necessarily occur by the nucleation of the product followed by its growth by the propagation of the interface between the parent and the product phases. The movement of the interface can be either thermally activated or athermal. The atom transfer process across the interface is thermally activated in the case of the former while it does not require the assistance of thermal fluctuations in the latter. The kinetic classification, originally introduced by Le Chatelier (Roy 1973), divides phase transformations into two main groups: (a) rapid or nonquenchable and (b) sluggish or quenchable. Transformations belonging to the former class are so fast that the parent phase, which is stable at high temperatures (or high pressures), cannot be retained by a rapid quench to ambient conditions. In contrast, sluggish transitions are slow to the extent that the high-temperature (or highpressure) phase can be retained metastably on quenching. The basic idea behind this classification scheme also centres around the requirement of thermal activation. This classification, however, suffers from the limitation that the experimental ability to rapidly change temperature and pressure is continuously improving and transformations which are grouped as “non-quenchable” today may become “quenchable” tomorrow. A truly displacive transformation occurs through the passage of a glissile interface which is essentially a displacement (or shear) front, the movement of which is not assisted by thermal activation. Such transformations are non-quenchable irrespective of the quenching rate employed.

2.4

SYNCRETIST CLASSIFICATION

The fundamental parameters on the basis of which a phase transformation is classified are thermodynamic, mechanistic and kinetic. A syncretist classification scheme has been introduced by Roy (1973) by taking all these aspects into account. Figure 2.10 shows a three-dimensional matrix with the x-, y- and z-axes representing the mechanistic (structural), thermodynamic and kinetic parameters, respectively. Along the x-axis the two major classes, namely, diffusional (reconstructive) and displacive transformations, are separated by a “mixed” class of transformations which have attributes of both displacive and diffusional transformations. Examples of each of these are available in Ti- and Zr-based systems. Martensitic transformations of the bcc () phase of pure Ti, Zr and of alloys based on these metals have been discussed in detail in Chapter 4 which also deals

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

106

5-6-2007

9:27 a.m.

Page:106

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

z

Thermally activated Athermal

Intermediate Rapid

Kinetic

Slow

y

ic

m

na

y od

rm

e Th

≠ t ΔV er rs Fi ord nd a ΔH

0

d ion ith V ixe it w , Δ M rans ect ΔH et ff all pr e sm

=0 d on der ΔV c d r Se o an

ΔH

Displacive

Mixed

Mechanistic (Structural)

Diffusional

x

Figure 2.10. Syncretist classification scheme of phase transformations based on mechanistic (structural), thermodynamic and kinetic criteria.

with martensites in NiTi-based intermetallics and ZrO2 -based ceramics. A host of diffusional transformations such as precipitation, amorphous-to-crystalline phase transformations, massive transformation, eutectoid phase reactions have also been encountered in these systems and they are discussed in Chapters 4 and 7. Displacive transformations can be further divided into different subgroups, depending on whether the transformation is dominated by lattice strains (martensitic transformation) or by shuffles (e.g. omega transition and ferroelectric transitions). The  →  transition which is frequently observed in several Ti- and Zr-based systems is unique with respect to lattice registry in three dimensions, pretransition effects and transformation product morphology. A detailed account of the -transformation is presented in Chapter 7. The classifications based on thermodynamic criteria, represented along the y-axis of Figure 2.10, divide phase transformations into first-order, higher-order and “mixed” transformations. The distinctions between these classes of transformations are illustrated in Figure 2.11 which depicts the variation of thermodynamic quantities such as specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature. In a first-order transition, there is a step change in these quantities at this temperature and there is no need for V , H and S of one phase to show a tendency to gradually approach the value corresponding to the other phase as the transition temperature is approached. A second-order transition is characterized by a gradual change in V , H and S and the absence of a step change in

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:107

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

107

Specific volume Enthalpy Entropy T

T

T

Temperature (a) First order

(b) Mixed

(c) Higher order

Figure 2.11. Changes in specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature in (a) second-order and (b) and (c) first-order transitions; (b) shows a “mixed” character, exhibiting pretransition effects as well as step changes at the transition temperature.

these parameters at the transition temperature. The mixed situation is encountered in a large number of transitions such as ferroelectric and ferromagnetic transitions. In such mixed transformations, though a pretransition second-order-like effect is observed, there is a finite discontinuous jump in the value of these thermodynamic quantities at the transition temperature. Considering kinetics as the third variable, phase transformations can be grouped into thermal and athermal classes. All true displacive transformations are athermal which cannot be suppressed by quenching. In contrast, reconstructive or diffusional transformations are invariably thermally activated and, therefore, such transformations are, in principle, suppressible on quenching. The required quenching rate for suppressing a diffusional transformation, however, varies depending on the incubation period involved. Having discussed different schemes of classification of phase transformations in alloys, intermetallics and ceramics, we will now examine how a given phase transformation can be classified on the basis of thermodynamic, kinetic and mechanistic criteria. A classification tree (Figure 2.12) can be constructed by addressing appropriate questions at different levels. The first question to raise is whether the transformation is homogeneous (or continuous) or discrete. Higher-order transitions are continuous while first-order transitions are discrete at the equilibrium transition temperature, Tc . Some firstorder transitions exhibit an instability temperature, Ti (Ti << Tc ), below which the transformation initiates in a continuous manner. Homogeneous transformations can further be subdivided depending on the nature of fluctuations (waves), the amplification of which represents the progress of the transformation, and the

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

108

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:108

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

Phase transformations (Thermodynamic) Higher order

First order Homogeneous continuous

Necessarily

Below instability temperature in selected cases

Discrete

Nucleation (Kinetic) Thermally activated

Athermal

Stable particles of new phase Growth (Mechanistic) Diffusional reconstructive

Long range

Displacive

Short range

Thermally activated atom transport

Bulk diffusion controlled

Interface diffusion controlled

Solute partitioning

Cellular Homogeneous precipitation and heterogeneous Eutectoid precipitation decomposition

Lattice strain dominated

Shuffle dominated

Cooperative movements of atoms, athermal

Interface diffusion controlled

Martensitic transformations

Displacive omega transformations

Composition invariant Composition invariant

Polymorphic crystallization Massive transformation

fcc → bcc, bct in Fe alloys

Chemical ordering

bcc → hcp in Ti, Zr alloys

bcc → ω in Ti, Zr alloys

Recrystallization

Figure 2.12. Classification tree for phase transformations.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:109

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

109

Table 2.1. Lattice wave descriptions of homogeneous transformations. Nature of wave

Long wavelength

Short wavelength

Replacive Displacive Electric Magnetic

Spinodal clustering Martensitic Ferroelectric Ferromagnetic

Spinodal ordering Omega Antiferroelectric Antiferromagnetic

wavelength associated with these waves. Table 2.1 lists different types of homogeneous transformations which can be distinguished on the basis of the nature of the wave and the wavelength. Discrete transformations, by definition, occur by the nucleation and growth process in which a small particle of the product phase (or a phase structurally and chemically close to the product phase) appears in the parent matrix. The two phases are separated by a sharp interface which gradually moves towards the parent phase, converting it to the product phase. Discrete transformations are subdivided into diffusional (or reconstructive) and displacive, based on the mechanism of atom transfer across the advancing transformation front. When the atom jump across the interface occurs by random diffusional jumps, the transformation is designated as a diffusional transformation. A subset of diffusional transformations is the replacive transformation which is accomplished by replacement of atoms in the lattice of the parent phase without destroying the parent lattice. Good examples of such transformations are those chemical ordering reactions which lead to the formation of a superlattice, without involving lattice reconstruction but rearrangement of atoms in the parent lattice. Phase transformations in which the chemical composition of the parent phase is not inherited by the product naturally involve partitioning of different components through long-range diffusion. Identification of such transformations as belonging to the category of diffusional transformations is rather trivial. Those cases where both the parent and the product phases have an identical composition, for example, the polymorphic transformation of a single-component system, the massive transformation of a multicomponent alloy or the martensitic transformation, can be grouped into the two categories, namely, displacive and diffusional, depending on the mechanism of atom transport across the transformation front. The mechanisms of atomic movements are difficult to observe through direct experiments. Therefore, the growth mechanism is inferred from a variety of crystallographic, morphological and kinetic observations such as presence of lattice correspondence, orientation relation, shape of the product phase, habit plane, macroscopic shape deformation, inheritance of atomic order and thermally activated or athermal nature of the growth process. These experimental observables

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

110

5-6-2007

9:27 a.m.

Page:110

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

are good indicators for deciphering the mechanisms of atomic movements and in a great majority of cases the growth mechanisms established from these observations have received universal acceptability. There are, however, some cases for which unanimity regarding the mechanism of atomic movements has not been arrived at. There is also a continued debate as to whether the operation of a shear mechanism in a transformation can be inferred from the existence of lattice correspondence and from the fact that the habit plane obeys the invariant plane strain condition. Some of these issues are discussed in later chapters of this book. The nature of the nucleation step in discrete transformations needs to be examined for recognizing the class of a given transformation. For example, a martensitic transformation can have either an athermal or a thermally activated nucleation step. In the former case, the overall transformation assumes an athermal character, while in the latter the volume fraction of the martensite grows with time sigmoidally at a constant temperature between the start (Ms ) and finish (Mf ) temperatures associated with martensite formation (Figure 2.13). Athermal nucleation is also encountered in diffusional transformations such as crystallization of metallic glasses. In this case, the nuclei of crystalline phases present in the asquenched amorphous matrix are activated for growth during the crystallization process. Diffusional transformations can be further subdivided into different categories based on the diffusion lengths of the different atomic species required to accomplish the transformation. In composition-invariant transformations, such as polymorphic transformations of single-component systems, polymorphic crystallization, massive transformation and partitionless solidification, the random

Isothermal

Athermal (Burst)

Athermal

T1

Martensite (%)

Ms

Martensite (%)

Martensite (%)

~100%

Ms > T1 > Mf

Mf Temperature (a)

Temperature

Time

(b)

(c)

Figure 2.13. (a) and (b) Volume fraction transformed as a function of temperature in the case of athermal nucleation of martensite. (c) Increase in volume fraction transformed with time at a constant temperature, T (Ms > T > Mf ), resulting from thermally activated nucleation of martensite.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:111

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

111

diffusive atom movements occur only across the advancing transformation front. The distances involved in the diffusion process are in the range of the nearest neighbour atomic distances. These transformations are, therefore, designated as short-range diffusional transformations. The partitioning of atomic species occurs between the product phases mainly by interfacial diffusion in cases such as eutectoid decomposition, eutectic crystallization and cellular precipitation, resulting in a two-phase lamellar product. The interlamellar spacing can vary from a few nanometres to a few micrometres. These transformations can be grouped into the category of intermediate-range diffusional transformations. Sometimes classification is made on the basis of the nature of diffusion – whether bulk or interfacial – which dominates in the transformation process. If one draws a comparison between the eutectic and the eutectoid decompositions, one can see that in the former case diffusion is mainly in the liquid phase ahead of the transformation front whereas in the latter case, partitioning of different components occurs primarily at the transformation front. As indicated earlier, displacive transformations are those which can be accomplished by introducing a lattice deformation in the parent lattice. In this class of transformations, a perfect lattice correspondence is maintained and chemical order, if present in the parent phase, is retained in the product phase. A very wide range of transformations are grouped in this class, which covers softmode ferroelectric and ferroelastic transitions in materials such as SrTiO3 and BaTiO3 , the omega transition in Ti and Zr alloys, shear transformations in  (bcc) phases in noble metal alloys and martensitic transformations in intermetallics such as nickel aluminides and nickel titanides, cubic-to-tetragonal or cubic-toorthorhombic transitions involving small lattice strains and classical systems of iron-based alloys. Since the characteristics of all of them are not the same, they have been further subdivided into different groups using different criteria for classification. In recent literature, Cohen et al. (1979), Delaye et al. (1982) and Christian (1990) have proposed these classification schemes which are summarized in Table 2.2. It has been mentioned earlier that a displacive transformation involves a homogeneous strain of the parent lattice which is accompanied by atomic shuffles (relative displacements of atoms within unit cells) in some cases. The relative contributions of the lattice strain and the shuffle can be used as important criteria for grouping martensites into two classes, namely, the shuffle dominated and the lattice strain dominated transformations. There are a number of examples of displacive transformations which exhibit pretransition softening of elastic moduli. Amongst these, Ni–Al alloys, containing 30–50% Al, constitute the most widely studied systems; they show weak to moderate first-order character. The fact that the high temperature 2 -phase

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

112

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:112

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

Table 2.2. Classification of displacive transformations. Criteria

Classification

Magnitude of shuffle and of homogeneous lattice strain (Cohen et al. 1979)

Presence of precursor mechanical instability (Delaye et al. 1982)

Structural basis (Christian 1990)

Shuffle dominated

Ferroelectric Ferromagnetic Omega

Lattice strain dominated

Martensitic: high strain energy controls morphology and kinetics Quasimartensitic: low strain energy; can occur continuously

No mechanical instability as Ms is approached Strongly first order

Allotropic changes in pure elements Transformations in primary solid solutions

Moderate indications of mechanical instabilities Moderate first order

-phases of noble metal alloys and Ni-based shape memory alloys

Marked mechanical instability Weakly first order, second order

Cubic → tetragonal Cubic → orthorhombic with small lattice strains

Between close packed layer structure

fcc, hcp, 9R, 18R, 4H, etc.; including monolithic and orthorhombic distortions

Between fcc, bcc and derived structure Between bcc, hcp and derived structure Between cubic and tetragonal and slight distortions

(CsCl structure) prepares itself for the transformation as the temperature is lowered towards the martensitic start (Ms ) temperature is well reflected in X-ray and electron scattering as well as in acoustic measurements. The entire 4 0 transverse acoustic phonon branch (corresponding to the shear modulus C  in the limit  → 0) is unusually low and the energy decreases considerably (though not to zero) at certain wave vectors as the temperature approaches Ms . This partial softening and the evolution of diffuse scattering due to elastic strain along the 0 directions indicate the existence of localized fine-scale displacement patterns. In the Ni625 Al375 alloy, the presence of localized displacements, which are remarkably similar to that required for the formation of the 7R martensite (the

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:113

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

113

stacking sequence of close packed planes being ABAC), has been observed in highresolution electron microscopy images just prior to the transformation (Tanner et al. 1990). As the parent phase approaches the Ms temperature, the density of such microregions deformed by lattice strains increases. The nucleation event in such cases can be viewed as a collapse or growth of microdomains associated with non-uniform lattice strains into a macrodomain of a size larger than the critical size of a nucleus and of homogeneous and nearly appropriate lattice strain. On the question of precursors in martensitic transformations, there are conflicting observations reported in different systems. Martensitic transformations can, therefore, be divided into different subclasses based on the nature and extent of precursor mechanical instability. Strongly first-order martensitic transformations do not show any softening of elastic constants as the system approaches the Ms temperature. In contrast, there are systems where moderate or marked indications of softening of elastic constants are present in the vicinity of the Ms temperature. Generally these “mixed” transformations, exhibiting partial mode softening, are associated with small lattice strains. Displacive-type structural transitions accompanying ferroelectric transitions in BaTiO3 can be cited as examples. These are associated with the displacement of a whole sublattice of ions of one type relative to another sublattice. The perovskite structure with a generalized composition ABO3 consists of a three-dimensionally linked network of BO6 octahedra, with A ions forming AO12 cuboctahedra to fill the spaces between BO6 octahedra. In view of these topological and geometrical constraints, there are only three structural degrees of freedom: (a) displacement of cations A and B from the centres of their cation coordination polyhedra, AO12 and BO6 , respectively; (b) distortions of the anionic polyhedra, coordinating A and B atoms; and (c) tilting of the BO6 octahedra about one, two or three axes. The first of these is the most important for the occurrence of ferroelectricity, since a separation of the centres of positive and negative charges corresponds to an electric dipole moment. Structural phase changes in BaTiO3 with temperature are shown in Figure 2.14. In cubic paraelectric BaTiO3 , both Ba and Ti have zero displacements, with perfectly regular polyhedra of coordinating oxygen ions. The tetragonal, monoclinic (orthorhombic) and rhombohedral forms, which are all ferroelectric, are associated with displacements of the ionic species and distortions of the polyhedra. In tetragonal BaTiO3 , both AO12 and BO6 are elongated along the c-axis, as c/a = 10098. Displacements of 9.68 pm for the Ba2+ ion and 11.50 pm for the Ti4+ ion along the tetragonal axis are responsible for creating a dipole moment with the polarization vector along the same direction. Smaller displacements of the oxygen ions contribute to distortions, with the four oxygen ions in the BO6 octahedron being perpendicular to the tetragonal axis displaced by 3.63 pm, in

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

114

5-6-2007

9:27 a.m.

Page:114

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys Cubic At 130°C; a 1 = a 2 = a 3 = 4.009 Å

Tetragonal

a2

a3

a1

130°C a 1 = a 2 = 3.992 Å

At 0°C

c

a2 a1

c = 4.035 Å Monoclinic a 1 = a 2 = 4.013 Å At c = 3.976 Å –90°C β = 98° 51′

0°C

a3

Rhombohedral

β a1

Orthorhombic c′

α

–90°C a3 a2

α

α b′

c

a′

a1

a ′ = 5.667 Å At b ′ = 5.681 Å –90°C c ′ = 3.989 Å

a 1 = a 2 = a 3 = 3.998 Å At –90°C α = 89° 52.5′

Figure 2.14. Sequence of phase transformations which occur in BaTiO3 at 130, 0 and −90 C. Unit cell dimensions and the orientation of the polarization vector are also indicated.

the opposite direction to the Ti4+ displacement. In the rhombohedral structure, displacements and distortions are correlated. In this case, displacements are parallel to the threefold axis which passes through two opposite triangular faces of the octahedron. The orthorhombic structure, by virtue of its lower symmetry, presents a wider range of polyhedral distortions. As is illustrated in Figure 2.14 the structure of BaTiO3 does not remain stable over the whole temperature range below the first ferroelectric Curie point and it transforms successively into lower symmetry variants, namely, cubic → tetragonal → monoclinic (orthorhombic) → rhombohedral. The vector directions of polarization are also indicated within the unit cells of these structures (Eric Cross 1993). Christian (1990) has grouped martensitic transformations in subcategories based on the structures of the parent and the product phases, as listed in Table 2.2.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:115

Trim:165 mm×240 mm

Classification of Phase Transformations

2.5

Integra, India

115

MIXED MODE TRANSFORMATIONS

The classification scheme discussed so far makes an attempt to assign a given transformation to a specific category based on thermodynamic, kinetic and mechanistic criteria. It must be emphasized that there exist several phase transformations in real systems which do not fall exclusively in a single category. These transformations, which exhibit characteristics of different classes of transformations, are often called “mixed mode” or “hybrid” transformations (Banerjee 1994). Some of these cases are briefly discussed here for the purpose of illustration. From thermodynamic considerations one can cite cases which show pretransition effects similar to those of second-order transitions and at the same time a sharp discontinuity of thermodynamic functions at the transition temperature. Though these have been designated as “mixed” in Figure 2.11 the sharp discontinuity at the transition temperature makes them first-order transitions as per the thermodynamic definition. Pretransition effects in these often arise due to quasistatic structural fluctuations (modulations in chemical composition or displacement) or modulations in the polarization of electric or magnetic vectors associated with lattice points. The interplay between more than one homogeneous phase transformations can be illustrated by (a) concomitant ordering and clustering processes and (b) sequential operation of spinodal clustering and magnetic ordering. 2.5.1 Clustering and ordering The formation of an ordered intermetallic phase from a supersaturated dilute solid solution often requires concomitant ordering and clustering. The conditions for either simultaneous or sequential operation of clustering and ordering processes have been identified (Kulkarni et al. 1985, Khachaturyan et al. 1988, Soffa and Laughlin 1989) in terms of instabilities associated with the clustering wave vector (k close to 000) and the ordering wave vector (k terminating at one of the special points of the reciprocal space). Let us consider an fcc solid solution which experiences the influences of 100 ordering instability and <000> clustering instability. The following situations can arise and the transformation sequence is accordingly selected: (a) The disordered solid solution is initially unstable with respect to 100 ordering but metastable against clustering. Ordering of the solid solution to an optimum level can introduce a tendency towards phase separation in the optimally ordered structure. (b) The disordered solid solution simultaneously exhibits 000 clustering and 100 ordering tendencies. Both the processes can proceed simultaneously, their relative kinetics determining the rates of progress of the two processes. (c) The peak instability temperature for 000 clustering is higher than that for 100 ordering. In this situation, clustering occurs first, creating solute-rich regions within which the ordering process sets in once the condition of ordering

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

116

5-6-2007

9:27 a.m.

Page:116

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

T1 αp

Tricritical point

αp

αf

Paramagnetic

Ferromagnetic P Q

A

Free energy

Temperature

Tc (X )

αf

R

Composition (%B)

A

Composition (%B) (b)

(a)

Figure 2.15. (a) A phase diagram showing a two-phase region introduced by a ferromagnetic ordering. (b) Free energy–concentration diagram at T1 showing the introduction of a spinodal clustering region in the ferromagnetic phase.

instability is fulfilled. Such coupled clustering–ordering processes are discussed in Chapter 7 in connection with phase transition sequences in Zr–Al and Cu–Ti alloys. Higher-order transitions like magnetic ordering can also induce a clustering tendency in a solid solution, resulting in the appearance of multicritical points in the phase diagram. Allen and Cahn (1982) have discussed these issues in great detail. Let us consider a binary fcc system of components A and B, where A is ferromagnetic and the Curie temperature, Tc (X), of the A–B solid solution changes with composition in the manner shown in Figure 2.15. In the absence of the magnetic contribution, the system behaves like an ideal solution while with the introduction of the magnetic contribution, the free energy of the -phase is reduced from that corresponding to paramagnetic p to that of ferromagnetic f . At temperatures below the tricritical point, a two-phase region emerges between f and p and two spinodes can also be identified in the free energy–composition plot (Figure 2.15(b)). At a temperature T1 an alloy at the point Q experiences a clustering instability and initially decomposes spinodally to two ferromagnetic phases, the one enriched in the non-magnetic component eventually undergoing a ferromagnetic → paramagnetic disordering. At the points P and R, the alloys are metastable with respect to spinodal clustering and, therefore, the paramagnetic phase, p , in the former and the ferromagnetic phase, f , in the latter can form only by nucleation and growth. 2.5.2 First-order and second-order ordering There are some instances where a system exhibits simultaneously a second-order and a first-order chemical ordering tendency. The ordering process in Ni4 Mo can

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:117

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

117

be cited in this context (Banerjee 1994, Arya et al. 2001). The Ni4 Mo alloy, quenched from the high-temperature disordered (fcc) phase field, exhibits (Spruiell and Stansbury 1965, Ruedl et al. 1968)  a short-range ordered (SRO) state characterized by diffraction intensity at 1 21 0 fcc positions and a complete   extinction of intensity at 210fcc positions in the reciprocal space. The 1 21 0 reflections do not coincide with the superlattice reflections of the equilibrium Ni4 Mo (D1a ) structure (Figure 2.16). While the SRO state consists of heterospace fluctuations   in the form of concentration wave packets of size 2–5 nm, with wave vectors 1 21 0 , the equilibrium D1a structure is associated with wave vectors, 15 420. The two competing superlattice structures, as shown in Figure 2.16(c) and (d), respectively, can be described in terms of 420 planes of all Ni (N) and all Mo (M) atoms in the stacking sequences of MMNNMMNN and MNNNNMNNNN, respectively. While  the 1 21 0 ordering fulfils all the symmetry criteria for a second-order transition, the 15 420 ordering is necessarily a first-order transition. In order to examine the relative strengths of the two  ordering tendencies, namely, the first order 15 420 and the second order 1 21 0 , the free energy (F ) surfaces, as functions of the respective order parameters,  1 21 0 and  15 420, have been calculated using first principles thermodynamic calculations (Arya et al. 2001). The instability of the system with respect to fluctuations corresponding to the two order parameters can be determined by examining the curvature ( 2 F/ 2 ) of the F versus  plots  1 1 at  = 0 along the two directions,  1 2 0 and  5 420. With decreasing temperatures, T1 , T2 , T3 and T4 , the following four situations arise, the corresponding free energy surfaces being depicted in Figure 2.17(a)–(d). (1) Tc (D1a ) < Tc (1 21 0) < T1 : positive curvatures for both 1 21 0 and 15 420 ordering, implying stability of the disordered state,  =  0.  (2) Tc (D1a ) < T2 < Tc (1 21 0): negative curvature for 1 21 0 and positive curvature  for 15 420 ordering, implying instability of the system for 1 21 0 ordering and no tendency towards D1a ordering.   (3) Ti (D1a ) < T3 < Tc (D1a ), < Tc (1 21 0): negative curvature for 1 21 0 and positive curvature for 15 420 ordering at  = 0 but a dip in the free energy with respect to the latter (D1a ordering) near  15 420 = 08. This  implies that the system 1 experiences simultaneously tendencies towards 1 2 0 ordering (second order) and D1a ordering (first order).   (4) T4 < Ti (D1a ) < Tc (D1a ) < Tc (1 21 0): negative curvatures along both 1 21 0 and 1 420 ordering implying 5  1  that 1the system is unstable with respect to the development of both 1 2 0 and 5 420 In this situation, homoge  ordering. neous ordering is feasible for both 1 21 0 and D1a ordering. A mixed   state consisting of concentration waves with wave vectors ranging from 1 21 0 to 1 420 is encountered on the path of the ordering process. This mixed state 5

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

118

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:118

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

(b)

(a)

Period

M

N N M M

N p: 0

1

2

3

4

N2 M 2 (420) -Unit Cell

N2 M2 -Tile

(c) M

M 4 N3 2 1 M

N p: 0 1

2 3

4

5

N4 M

(420)

N4 M

(d)

Figure 2.16. Electron  diffraction patterns corresponding to (a) the “short-range ordered” structure characterized by 1 21 0 reflections and complete extinction of 210 reflections and (b) the D1a ordered structure in the Ni–25 at.% Mo alloy. Real lattice descriptions of the fcc-based superstructures in terms of stacking of 420 planes in the [001] projections and static concentration waves are  shown in (c) for 1 21 0 ordering and in (d) for D1a -Ni4 Mo ordering with wave vector 15 420. The sequences of Ni (N) and Mo (M) layers of (420) planes and subunit cell clusters are also shown.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Trim:165 mm×240 mm

Page:119

Classification of Phase Transformations

119

6

8

T1 > T2 > T3 > T4

6

4

4

F ord (K)

Integra, India

T1

2

2

T2

T1

0

T2 0

ηc

–2

T3

ηc

T3

–2

–4

T4

T4

–4 0.0

0.2

0.4

0.6

0.8

1.0

–6 0.0

0.2

0.4

0.6

0.8

1.0

η /ηmax   Figure 2.17. The ordering free energy of the Ni–25 at.% Mo alloy, exhibiting the 1 21 0 and the 1 420 ordering tendencies, plotted as a function of order parameters for the corresponding ordering 5 wave vectors at four different temperatures, T1 , T2 , T3 and T4 , respectively, pertinent to the situations described in the text.

is characterized by diffraction (Figure 2.18) which show a spread of  patterns  diffracted intensity linking 1 21 0 and 15 420 positions and by the presence of  subunit cell clusters (or motifs) representing 1 21 0 and 15 420 ordered structures (as illustrated in Figure 2.16) in lattice resolution images (Figure 2.18).

(a)

(b)

  Figure 2.18. Microstructure and diffraction pattern corresponding to mixed 1 21 0 and 15 420 ordering: (a) diffraction pattern showing intensity distribution linking 1 21 0 and 15 420 spots; (b) high-resolution electron micrograph showing motifs of D1a and N2 M2 structures (as schematically illustrated in Fig. 2.16 (c) and (d)).

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

120

5-6-2007

9:27 a.m.

Page:120

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

2.5.3 Displacive and diffusional transformations Phase transformations are classified as displacive and diffusional on the basis of the nature of atom movements across the advancing transformation front. One can envisage transformations which occur by a coupling between a displacive and a diffusional mode of atomic movements. The formation of ordered -structures from the disordered parent bcc -phase can be cited as an appropriate example of a mixed diffusive/displacive transformation in which the bcc lattice is transformed into the hexagonal -structure by a periodic displacement of lattice planes while the decoration of the -lattice by different atomic species occurs through diffusional atom movements. These two processes can as well be designated as displacive and replacive ordering, respectively, and the overall process can be viewed as a superimposition of a displacement wave and a concentration wave on the bcc lattice (Banerjee et al. 1997). This mode of transformation (discussed in Chapter 6) is encountered in several bcc Ti and Zr alloys, leading to the formation of a wide variety of ordered -structures. Displacive and diffusional atom movements can also be coupled through kinetic considerations in several cases, one of the best examples being the formation of -hydride precipitates in either the - or the -phase matrix alloys – a topic discussed in detail in Chapter 8. The formation of the -hydride phase from either the - or the -phase involves a shear transformation of the parent lattice accompanied by partitioning of hydrogen atoms (discussed in Chapter 8). The latter process being exceptionally rapid, the displacive lattice shear and hydrogen partitioning can occur nearly concurrently. Hydride formation in Zr- and Ti-based alloys can be compared with bainite formation in steels. 2.5.4 Kinetic coupling of diffusional and displacive transformations Olson et al. (1989) have analysed the kinetics of a transformation process in which the product phase forms with a partial redistribution of the interstitial element during non-equilibrium nucleation and growth. The rate at which the advancing transformation front moves depends both on its intrinsic mobility and on the ease with which the interstitial element diffuses ahead of the moving interface. The intrinsic mobility is related to the process of structural change across the moving interface. The growth, involving partial supersaturation in which local equilibrium is not established at the interface, seems to be an unstable process. This is because a perturbation in composition towards equilibrium would lead to a reduction in free energy which would drive the system to attain a local equilibrium. Stability of the non-equilibrium growth mechanism can, however, be brought about by another process, such as structural transformation across the interface, occurring in series. A displacive structural transition involving movement of a glissile interface and

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:121

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

121

Xl

γ

Gdd

G id

Carbon →

Free energy

α

γ

X

Xα

Xα

Distance →

Xl Xm

X

Carbon concentration

(b)

(a)

Figure 2.19. (a) Free energy–concentration plots for ferrite () and austenite () showing the free energy component dissipated in structural change at the interface, Gid , and the free energy component dissipated in diffusive movements of interstitial atoms, Gdd , ahead of the transformation front; (b) shows concentration distribution in the - and -phases.

diffusive movements of interstitial atoms ahead of the interface are, therefore, modelled as coupled processes resulting in the bainitic transformation in steels. Both these processes dissipate the net free energy, G (as indicated in Figure 2.19) which is made up of Gid and Gdd amounts dissipated in the interface process and the diffusion process, respectively. The interfacial velocities can be calculated for the two processes and expressed as Vi =  Gid and Vd =  Gdd

(2.6)

where  and  are response functions relating the velocity to the appropriate dissipation. For the process to be kinetically coupled, the interface velocity, V = Vi = Vd . The interface velocity is calculated on the basis of thermally activated motion of dislocations which constitute the ferrite/ austenite glissile interface and is found to be comparable with the diffusion field velocities computed for different levels of interstitial supersaturation. It may be noted that the two types of thermally activated events which are coupled in this treatment operate on widely differing size scales. The manner in which the unit processes of the displacive and the diffusional aspects of the transformation couple at the microscopic level can be understood in terms of the discontinuous nature of the thermally activated interfacial motion. During the “waiting time” prior to a thermally activated event, the solute partitioning in the vicinity of the interface can cause a steady increase in the local interfacial driving force till it reaches a threshold value where the interface is driven to its next position of temporary halt. The movement between these positions occurs through a diffusionless “free glide” motion.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

122

5-6-2007

9:27 a.m.

Page:122

Trim:165 mm×240 mm

Integra, India

Phase Transformations: Titanium and Zirconium Alloys

This coupled transformation model has been applied to bainitic transformations in steels. The model predicts an increasing interfacial velocity and supersaturation during growth with decreasing transformation temperature, while the nucleation velocity passes through a maximum, giving C-curve kinetics. In this context, it should be mentioned that there is a different view point on the mechanism of the bainitic transformation in which atom transport across the interface is considered to occur through diffusional random jumps. The displacive process, involving coordinated atomic jumps from a parent lattice site to a predestined site in the product lattice, has not been accepted as a requirement for a bainitic transformation in the diffusionist view which has been summarized in an excellent manner by Reynolds et al. (1991).

REFERENCES Allen, S.M. and Cahn, J.W. (1982) Bull. Alloy Phase Diagrams, 3, 287. Arya, A., Banerjee, S., Das, G.P., Dasgupta, I., Saha-Dasgupta, T. and Mookerjee, A. (2001) Acta Mater., 49, 3575. Banerjee, S. (1994) Solid → Solid Phase Transformations (eds W.C. Johnson, J.M. Howe, D.E. Laughlin and W.A. Soffa) TMS, Warrendale, PA, p. 861. Banerjee, S., Tewari, R. and Mukhopadhyay, P. (1997) Prog. Mater., Sci., 42 (1–4), 109. Buerger, M.J. (1951) Phase Transformations in Solids, Wiley, New York, p. 183. Christian, J.W. (1965) Physical properties of martensite and bainite, Special Report 93, Iron and Steel Institute, London, p. 1. Christian, J.W. (1979) Phase Transformations, Vol. 1, Institute of Metallurgists, London, p. 1. Christian, J.W. (1990) Mater. Sci. Eng., A127, 215. Cohen, M., Olson, G.B. and Clapp, P.C. (1979) Proceedings of International Conference on Martensite, MIT Press, Cambridge, MA, p. 1. Cook, H.E. (1974) Acta Metall., 22, 239. de Fontaine, D. (1975) Acta Metall., 23, 553. Delaye, L., Chandrasekaran, M., Andrade, M. and Van Humbeck, J. (1982) Solid– Solid Phase Transformations (eds H.I. Aaronson, D.E. Laughlin, R.F. Sekerka and C.A. Wayman) TMS, Warrendale, PA, p. 1429. Ehrenfest, P. (1933) Proc. Acad. Sci. Amst., 36, 153. Eric Cross, L. (1993) Ferroelectric Ceramics., (eds Nava Setter, Enrico L. Colla and Birkhaeuser Basel), Switzerland. Khachaturyan, A.G., Lindsey, T.F. and Morris, J.W. (1988) Metall. Trans., 19A, 249. Kulkarni, U.D., Banerjee, S. and Krishnan, R. (1985) Mater. Sci. Forum, 3, 111. Landau, L.D. and Lifshitz, E.M. (1969) Statistical Physics, Pergamon Press, Oxford. Olson, G.B., Bhadeshia, H.K.D.H. and Cohen, M. (1989) Acta Metall., 37, 381. Rao, C.N.R. and Rao, K.G. (1978) Phase Transitions in Solids, McGraw Hill, New York.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto

Elsevier UK

Code: PTA

Ch02-I042145

5-6-2007

9:27 a.m.

Page:123

Trim:165 mm×240 mm

Classification of Phase Transformations

Integra, India

123

Reynolds, W.T., Jr, Aaronson, H.I. and Spanos, G. (1991) Mater. Trans. JIM, 32, 737. Roy, R. (1973) Phase Transitions (ed. L.E. Cross) Pergamon Press, Oxford, p. 13. Ruedl, E., Delavignette, P. and Amelinckx, S. (1968) Phys. Status Solidi, 28, 305. Soffa, W.A. and Laughlin, D.E. (1989) Acta Metall., 37, 3019. Spruiell, J.E. and Stansbury, E.E. (1965) J. Phys. Chem. Solids, 26, 811. Tanner, L.E., Schryvers, D. and Shapiro, S.M. (1990) Mater. Sci. Engi., A127, 205.

Font:Times

F.Size:11/13 pt

Margins: Top:19 mm

Gutter:19 mm

Width:128 mm

Depth:40 Lines

1 Color

Recto