Classification of Phase Transformations

Functions and discontinuity in differentials f(x) g(x) h(x)

g(x) h(x) f(x) Discontinuity in the curvature Discontinuity in the slope g(x) Discontinuity in the function h(x) f(x)

Classification of Phase Transformations Order of a phase transformation Thermodynamics Ehrenfest, 1933 Classification of Phase Transformations B Based on Mechanism Mechanistic Buerger, 1951 C Kinetics le Chatelier, (Roy 1973)

B Reconstructive Mechanistic Displacive Diffusional → Civilian Subset Breaking of bonds and formation of new ones Atom movements from parent to product by diffusional jumps Nearest neighbour bonds broken at the transformation front and the product structure is reconstructed by placing the incoming atoms in correct positions → growth of product lattice Diffusional transformation  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 Homogenous distortion or a combination E.g.: Martensitic Formation of nucleus of product Movement of shear front at speed of sound Shuffling of lattice planes Static displacement wave

Displacive 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").

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

Buerger’s classification: full list 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)

C Non-quenchable Kinetic Quenchable Athermal → Rapid 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

Thermodynamic classification Ehrenfest, 1933 Order of a phase transformation The lowest derivative (n) which shows a discontinuity at the transition point Can be used for equilibrium transitions of single component systems There are cases of mixed order transformations In thermal transformations: usually the high-T form is of higher symmetry and higher disorder

n = 1 Finite discontinuity First Order 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)

Schematics

n = 2 Finite discontinuity Second Order 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

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

Schematics In a two component system: 1st order transformation appears in a phase diagram as two line 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

n = 3 Third Order 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

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)

n = 1 Not zero Note the barrier Two minima separated by a G barrier First Order 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

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

n = 2 Second Order 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

 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

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

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

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 xamples 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

Inverted image (black → white) looks very similar! A coarsened spinodal microstructure in Al-22.5 at.% Zn-0.1 at.% Mg solution treated 2h at 400C and aged 20h at 100C. TEM micrograph at 314 kX. (K.B. Rundman, Metals Handbook, 8th edn. Vol.8, ASM, 1973, p.184.

Nucleation & Growth Spinodal The composition of the second phase remains unaltered with time A continuous change of composition occurs until the equilibrium values are attained The interfaces between the nucleating phase and the matrix is sharp The interface is initially very diffuse but eventually sharpens There is a marked tendency for random distribution of both sizes and positions of the equilibrium phases A regularity- though not simple- exists both in sizes and distribution of phases Particles of separated phases tend to be spherical with low connectivity The separated phases are generally non-spherical and posses a high degree of connectivity

Spinodal Ordering Ordering leads to the formation of a superlattice Ordering can take place in Second Order or First 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

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 Tm G → Liquid stable Solid stable Solid Metastable Liquid Metastable For a second order transformation the free energy curve is a single continuous curve and the concept of a metastable state does not exist

Enantiotropic transformations Equilibrium 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 transformations Irreversible (no equilibrium between parent and product phases) A’ (metastable) → B (stable) (at T1) Supercooled liquid (metastable) → A (stable) (at T2)

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

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

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

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