Classification of Phase Transformations

1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm Classification of Phase Transformations

2. Mathematical basics • Let us start with some mathematical basics. This is required as in the thermodynamic classification we will deal with discontinuity of functions. • Either the function itself may have a discontinuity or one of its derivatives. Usually the first few derivatives have a ‘physical meaning’ as well. • As we shall see in the first order transformation the plot of enthalpy of the transformation versus T shows a discontinuity, while in second order transformations the derivative of enthalpy with T (i.e. CP) shows a discontinuity. • The important point to note that this thermodynamic basis of classification (based on order) can be correlated with ‘microstructural processes’ leading to the transformation. • A first order transformation like solidification of a pure metal (e.g. Cu) takes place by nucleation and growth. In a second order (2) transformation (e.g. order-disorder transformation in -brass) the entire volume transforms. • In nucleation (& growth) a ‘completely new state’ is created, which then grows at the expense of the parent state. In a second order transformation (like ordering in some systems), order slowly evolves in the entire volume. Another example of a 2 transformation is ferromagnetic to paramagnetic transformation in some systems.

3. Functions and discontinuity in differentials • Here we have taken a step function f(x) with a discontinuity at x = 0. By integrating it we get a continuous function g(x). • By further integrating it we get function h(x). f(x) • Let us consider a function h(x). • The curvature of this function h(x), which is f(x) has a discontinuity. • The slope of this function g(x) is continuous, but its slope, which is f(x) has a discontinuity. g(x) h(x)

4. Now let us plot these functions Discontinuity in the curvature Discontinuity in the slope g(x) Discontinuity in the function h(x) f(x)

5. Classification of Phase Transformations • Traditionally phase transformations have be classified based on:  Thermodynamics (due to Ehrenfest, 1933),  Mechanism (due to Buerger, 1951),  Kinetics (le Chatelier, (Roy 1973). A Order of a phase transformation Thermodynamics Ehrenfest, 1933 B Classification of Phase Transformations Based on Mechanistic Mechanism Buerger, 1951 C le Chatelier, (Roy 1973) Kinetics

6. A Thermodynamic classification Order of a phase transformation Ehrenfest, 1933 • The thermodynamic classification can be used for equilibrium transitions of single component systems. • The derivative of a thermodynamic variable like G is obtained with T. The order of the transformation is the lowest derivative (n) which shows a discontinuity at the transition point (i.e. T = TC). • ‘n’ can take values like 1, 2, … , which correspond to First order (1), Second order (2), etc. transformation. • Usually, one does not consider transformations of order greater than 2. There are cases of mixed order transformations. • In thermal transformations: usually the high-T form is of higher symmetry and higher disorder. The lowest derivative (n) which shows a discontinuity at the transition point

7. n = 1 First Order Finite discontinuity Schematics • First order transitions are characterized by discontinuous changes in entropy, enthalpy & specific volume. • H → change in enthalpy corresponds to the evolution of Latent Heat of transformation • The specific heat [J/K/mole] is thus infinite (i.e. at the transition heat is being put into the system but the temperature is not changing)

8. Funda Check How do I understand a first order phase transformation? • In a first order transformation the product phase (which first forms a a nucleus) is very different from the parent phase in some way (e.g. in solidification the crystal is ordered while the liquid is not). • During solidification the enthalpy of fusion is released and hence the enthalpy of the solid is different from that of the melt (liquid). Hence, the H vs T curve shows a discontinuity at the melting point (TM). • Similarly, the entropy of the parent and product phases are very different (liquid has high entropy while the solid has low entropy) and a plot of S vs T curve will show a discontinuity.

9. Finite discontinuity Second Order n = 2 Second derivative is CP • NO discontinuous changes in entropy, enthalpy & specific volume. • NO latent heat of transformation. • High specific heat at the transition temperature. • Finite discontinuity in CP(NOT infinite). • Lamda () Transitions (-point transitions) show infinity. • Concept of a metastable phase not readily applicable to a 2 transition → single continuous free energy curve. • Ferromagnetic ordering, Chemical ordering are examples of 2 transitions. • In a two component system a 2nd order transformation requires equality of entropy and volume of two phases + identical composition of the two phases. Quartz

10. 2 transitions can be described by mean field descriptions of cooperative phenomenon. • Order parameter continuously decreases to zero as T → TC. • Any transition which can be described by a continuous change in one or more order parameters can be treated by a the generalized LANDAU Equation.

11. Funda Check In a second order transformation what is meant by ‘the whole volume transforms’?

12. Schematics In a two component system: • 1st order transformation appears in a phase diagram as two lines bounding the region where two phases (of different composition) coexist. • Second order transformation appears as a single line. Phase Transformations: Examples from Ti and Zr Alloys, S. Banerjee and P. Mukhopadhyay, Elsevier, Oxford, 2007

13. Third Order n = 3 • There is usually no classification as third order (II and higher order are clubbed together). • Superconducting transition in tin at zero field & Curie points in many ferromagnets can be considered as third order transitions. Mixed Order

14. From the mechanistic view point two broad classes exist:  Reconstructive and  Displacive. • In reconstructive transformations ‘breaking of bonds’ takes place and formation of new ones. Here, atom movements from parent to product phase takes place by diffusional jumps. Nearest neighbour bonds are broken at the transformation front and the product structure is reconstructed by placing the incoming atoms in correct positions → leading to the growth of product lattice. • In Displacive transformations (e.g. Martensitic transformation), there is a cooperative motion of a large number of atoms (i.e. bonds are not ‘broken’ in the usual sense). • In diffusional transformations  Even in case chemical composition same (parent & product) + strict orientation relation → Still lattice correspondence not present • E.g.: Precipitation in Al-Cu alloys • Nucleation of product • Growth Reconstructive Diffusional → Civilian B Subset Replacive → e.g. ordering Mechanistic Buerger, 1951 Displacive Military Cooperative motion of a large number of atoms • E.g.: Martensitic • Formation of nucleus of product • Movement of shear front at speed of sound or a combination Homogenous distortion Shuffling of lattice planes Static displacement wave

15. Shuffle dominated Lattice strain dominated Magnitude of shuffle and of homogenous lattice strain Presence of precursor mechanical instability Structural basis Displacive Homogenous distortion Shuffling of lattice planes Static displacement wave • Coordination between neighbours retained in the product lattice (though bond angles change) • Atomistic coordination inherited → chemical order in parent structure is fully retained in the product structure similar correspondence of crystallographic planes Lines → lines; planes → planes (vector, plane, unit cell correspondence)  AFFINE TRANSFORMATION In general, an affine transform is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift").

16. NOTE: Lattice correspondence does NOT imply ORIENTATION RELATION  as phase transformations may involve rigid body rotations

17. Buerger’s classification: full list • Buerger had given an classifcation of phase transformations as listed below. • Transformation involving first coordination Reconstructive (sluggish) Dilatational (rapid). • Transformation involving second coordination Reconstructive (sluggish) Displacive (rapid). • Transformations involving disorder  Substitutional (sluggish) Rotational (rapid). • Transformations involving bond type (sluggish).

18. Classification based on Kinetics • Some transformations are aided by thermal activation (e.g. transformations involving diffusion). The kinetics of such transformations can be lowered by decreasing the temperature. As ‘sufficiently low’ temperature, the process can be considered as ‘frozen’ or quenched. This low temperature has to be obtained very fast (by fast cooling) and hence this class of transformations are considered as quenchable. • On the other hand those transformations which do not require thermal activation (athermal) cannot be ‘frozen’ by lowering the temperature. Such transformations (e.g. martensitic transformation*) are non-quenchable. Non-quenchable Athermal → Rapid C Subset Replacive → e.g. ordering Kinetic le Chatelier, (Roy 1973) Quenchable Thermal → Sluggish * Usually martensitic transformations are athermal- however there are instances of they being isothermal.

19. Landau Equation • Close to the critical temperature:The free energy difference (G) between finite and zero values of order parameter () may be expanded as power series. • Practically, any physical observable quantity which varies with temperature (or other thermodynamic variable) can be taken as a experimental order parameter. A, B, C.. = f(T, P)

20. First Order n = 1 Not zero Note the barrier • Two minima separated by a G barrier • T slightly less than TC the system still not unstable at  = 0 (state) (curvature remains +ve) a gradual transition of the system in a homogenous fashion to a the free energy minimum at  = C (or near it) is not possible

21. Phase transition can initiate if localized regions are activated to cross the free energy barrier (beyond  = *) → where phase with finite  can grow spontaneously • Formation of localized product phase regions with  ~ C → nucleation Sharp interface between parent and product phases Nucleation and Growth  Discrete nature of the transformation

22. Second Order n = 2 • Only even powers • Single equilibrium at  = 0 → corresponds to +ve value of A • +ve curvature • Curvature at  = 0 decreases • System becomes unstable at T = TC and fluctuations will lead to lowering of energy • ve curvature at  = 0→ corresponds to ve value of A • Glass transitions, Paramagnetic-Ferromagnetic transitions

23. Lambda transitions  • Heat capacity tends to infinity as the transformation temperature is approached • E.g.: Transformation in crystalline quartz Order-disorder transition in -brass (B2 → BCC, Cu-Zn alloy) • Symmetrical -transition → Manganese Bromide Manganese Bromide: Symmetrical -transitions Quartz: Unsymmetrical -transitions

24. Homogenous (Continuous) Transitions • Parent phase gradually evolves into the product phase without creating a localized sharp change in the thermodynamic properties and structure in any part of the system • The system becomes unstable with respect to small (infinitesimal) fluctuations → leading to the transition • The free energy of the system continuously decreases with amplification of such fluctuations • USUALLY • First order transitions are discrete (For T > Ti )→ Nucleation and Growth • Higher order transitions are homogenous→ parent and product phase cannot be sharply demarcated at any stage of the transition NO Sharp interface between parent and product phases Continuous nature of the transformation

25. For T < Ti • First order transitions are can proceed in a continuous mode • Not all first order transitions have a instability temperature • Examples of first order continuous transitions (conditions far from equilibrium):Spinodal clusteringSpinodal orderingDisplacement ordering

26. Spinodal clustering Spinodal decomposition • Phase diagrams showing miscibility gap correspond to solid solutions which exhibit clustering tendency. • Within the miscibility gap the decomposition can take place by either:  Nucleation and Growth (First order) or by  Spinodal Mechanism (First order). • If the second phase is not coherent with the parent then the region of the spinodal is called the chemical spinodal. • If the second phase is coherent with the parent phase then the spinodal mechanism is operative only inside the coherent spinodal domain. • As coherent second phases cost additional strain energy to produce (as compared to a incoherent second phase – only interfacial energy involved) → this requires additional undercooling for it to occur. • Spinodal decomposition is not limited to systems containing a miscibility gap. • Other examples are in binary solid solutions and glasses. • All systems in which GP zones form (e.g.) contain a metastable coherent miscibility gap → THE GP ZONE SOLVUS. • Thus at high supersaturations it is GP zones can form by spinodal mechanism.

27. Inverted image (black → white) looks very similar! A coarsened spinodal microstructure in Al28Cr27Fe25Ni20 alloy.

31. Spinodal Ordering • Ordering leads to the formation of a superlattice • Ordering can take place in Second Order orFirst Order (in continuous mode below Ti) modes • Any change in the lattice dimensions due to ordering introduces a third order term in the Landau equation • Continuous ordering as a first order transformation requires a finite supercooling below the Coherent Phase Boundary to the Coherent Instability (Ti) boundary • These (continuous ordering) 1st order transitions are possible in cases where the symmetry elements of the ordered structure form a subset of the parent disordered structure Not zero

33. Liquid Metastable Solid Metastable G → Solid stable Liquid stable Tm METASTABLE STATE For a first order transformation the free energy curve can be extrapolated (beyond the stability of the phase) to obtain a G curve for the metastable state For a second order transformation the free energy curve is a single continuous curve and the concept of a metastable state does not exist

34. Enantiotropic transformationsEquilibrium transitions: Reversible and governed by classical thermodynamics • L → A (at the melting point: Tm = TL/A) • A → B (at the equilibrium transformation T: TL/A) • A → A’ (transformation between two metastable phases) Monotropic transformationsIrreversible (no equilibrium between parent and product phases) • A’ (metastable) → B (stable) (at T1) • Supercooled liquid(metastable) → A (stable) (at T2)

35. Displacive transformation • Changes in higher coordination effected by a distortion of the primary bond • Smaller changes in energy • Usually Fast • High temperature form → more open, higher specific volume, specific heat, symmetry • E.g.:  high-low transformations of quartz (843K), tridymite (433K & 378K), cristobalite (523K)  SrTiO3

36. Toy Model for Displacive Transformation M.J. Buerger, Phase Transformations in Solids, John Wiley, 1951

37. L +  L  L +  L Ordered solid ’ A B   + ’ Ordered solid ’ A B  + ’

38. A B L L +  E.g. Au-Ni  2 1 1 + 2