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Basics of Phases and Phase Transformations W. Püschl University of Vienna

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Content 1. Historical context 2. Classification of phase transformations 3. Graphical thermodynamics – Phase diagrams 4. Miscibility Gap – Precipitation nucleation vs. spinodal decomposition 5. Order 6. Ising model: atomic and magnetic spin configuration 7. Martensitic transformations

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Early technological application of poly-phase systems: Damascus Steel

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Aloys v. Widmannstätten 1808 Iron meteorite cut, polished, and etched: Intricate pattern appears

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Oldest age hardening curve: Wilms Al-Cu(Mg,Mn,Fe, Si) alloy Retarded precipitation of a disperse phase.

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A scientific understanding of phases and phase transformation begins to develop end 19th / beginning 20th centuries physical metallurgy Experimental: Gustav Tammann (Göttingen) Theoretical: Josiah Willard Gibbs

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What is a phase? Region where intrinsic parameters have (more or less) the same value lattice structure, composition x, degree of order , density ,… Need not be simply (singly) connected. Expreme example: disperse phase and matrix phase where it is embedded (like Swiss cheese) When is a phase thermodynamically stable? How can we determine wihich phase is stable at a certain composition, temperature (and pressure, magnetic field…) What happens if this is not the case metastability or phase transition How can a phase transition take place?

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Ehrenfest (1933) 1st oder phase transition2nd order (generally: higher order)

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Free energy vs. order parameter according to Landau Higher-order phase transition 1st oder phase transition

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Chemical potentials g i of the components

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Gibbs phase rule f = (n - 1) – n ( - 1) + 2 = n - + 2

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Liquid-solid transition of a two-component System (Ge-Si)

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Excess enthalpy and miscibility gap

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Excess enthalpy and miscibility gap

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Precipitation: alternative mechanisms

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Heterophase fluctuation corresponds to nucleation Homophase fluctuation corresponds to spinodal decomposition

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Free energy of a spherical precipitate particle

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Ni 36 Cu 9 Al 55 Precipitation by nucleation and growth: N V particle number, c supersaturation, mean particle radius

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Spinodal Decomposition

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Excess enthalpy Positive: like atoms preferred: Phase separation Negative: unlike atoms preferred: ordering Short range order: there is (local) pair correlation Cowley- Warren SRO parameter Decay with distance from reference atom If they do not decay long range order

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disordered state (bcc) Long range order: out of bcc structure the B2 (CsCl) structure arises ordered state (B2) disordered state (bcc)

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Long range order: B2 (CsCl) representatives

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D0 3 stoichiometry 3:1 Out of the same bcc structure:

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L1 2 Long range order out of the fcc structure:

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L1 2 ordered state L1 2 disordered state (fcc)

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L1 2 representatives

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fcc L1 0 stoichiometry 1:1

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representatives fcc L1 0 stoichiometry 1:1

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Different long range ordered structures in the Cu-Au phase diagram L1 2 L1 0 L1 2 L1 0

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L1 2 L1 0 CuAu II (long period.) L1 2 L1 0 Different long range ordered structures in the Cu-Au phase diagram

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Fcc L1 1 stoichiometry 1:1

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Statistical physics of ordered alloys Partition function Possibly different vibration spectrum for every atom configuration Does it really matter?

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FePd: Density of phonon states g( ) L1 0 - ordered fcc disordered Mehaddene et al. 2004

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Bragg – Williams model: only nn pair interactions, disregard pair correlations R long range order parameter tanh R/ <1 >1 Simplifying almost everything:

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Different levels of approximation in calculating internal interaction energy Bragg-Williams Experiment Quasi-chemical quasi-chemical Experiment Bragg-Williams

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Ising model (Lenz + Ising 1925) Can be brought to Ising form by identifying (for nn interaction) Hamiltonian for alloy (pair interaction model) p i n atom occupation function

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Idea of mean field model: treat a few local interactions explicitly, environment of similar cells is averaged and exerts a mean field of interaction

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Local interaction only 1 atom Bragg- Williams – model Correspondences: Phase-separating ----- ferromagnetic Long range ordering ----- antiferromagnetic ferromagnetic

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Structure on polished surface after martensitic transformation: roof-like, but no steps. A scratched line remains continuous

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Martensite morphologies

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Homogeneous distorsion by a martensitic transformation

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First step :Transformation into a new lattice type: Bain transformation

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Second step: Misfit is accomodated by a complementary transformation: twinning or dislocation glide

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Thermoelastic Martensites: Four symmetric variants per glide plane: Can be transformed into one another by twinning

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Shape Memory effect

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Final remarks: As the number of components grows and interaction mechanisma are added, phase transformations can gain considerable complexity For instance: Phase separation and ordering (opposites in simple systems) may happen at the same time. I have completely omitted many interesting topics, for instance Gas-to-liquid or liquid-to-liquid transformations The role of quantum phenomena at low-temperature phases Dynamical phase transformations, self-organized phases far from equilibrium

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