lectures29-31 - Kinetics of Phase Transformations...

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Unformatted text preview: Kinetics of Phase Transformations (References: J. W. Christian, Theory of Transformations in Metals and Alloys; Porter and Easterling, Phase Transformations in Metals and Alloys) Phase Transformations Change of state, e.g. s l v l l s , , Change of structures, e.g. T C Change of composition, e.g. phase separation Why does a phase transformation take place? Thermodynamics How does a phase transformation take place? Kinetics Types of transformations diffusional diffusionless nucleation and growth Continuous (spinodal) nucleation and growth Continuous (spinodal) Examples of diffusional transformations: order-disorder, phase separation, precipitation, solidification in alloys, etc. Examples of diffusionless transformations: Martensitic transformations, structural transformations without compositional changes such as cubic to tetragonal transformations in BaTiO 3 . Cahn-Hilliard Diffusion Equation and Spinodal Decomposition Diffuse-Interface Theory Equilibrium Composition Profile Interfacial Free Energy of a Flat Interface Interfacial Width Up-Hill Diffusion Cahn-Hilliard Diffusion Kinetics of Spinodal Decomposition o X ' X ' ' X Phase Separation T X " ' + Monte-Carlo Simulation of phase separation Microscopic diffusion equation simulation of phase separation Examples : liquids, glasses, polymer blends, solids such as Al-Zn, TiO 2-SnO 2 , Al-Cu, CoO-MgO, etc. ( ) ( ) ( ) B B B v X f X f dx A dx X f A F + = = + + v Its only a function of volume fraction Free Energy of a Two-Phase Mixture ( ) x c B x c c two phase The free energy of a two- phase mixture with equilibrium compositions e unit volum per mixture phase- two of energy free v F phase of fraction volume phase of fraction volume e unit volum per phase of energy free f e unit volum per phase of energy free f n compositio B X area sectional- cross A The Diffuse-Interface Theory ( ) ( ) [ ] + = v 2 v v 2 1 dV X X f V F B B In general, the local free energy depends not only the local composition but also the composition of the immediate environment. In the diffuse-interface description, the local free energy is expressed as a function of the local composition and the local composition gradient, respectively, i.e. where is called the gradient energy coefficient and the corresponding term is called gradient energy homogeneous free energy free energy due to inhomogeneity Variational Derivative ( ) dx x X X f F B B + = 2 2 ( ) 2 2 x X X X f X F B B B B = ( ) ( ) = dx dx dy x y x I Y , , ( ) ( ) dx dy I dx d y I x y Y = y ( x ) and dy / dx are two independent variables For example Chemical Potential in Inhomogeneous Systems ( ) 2 2 x X X X f X F B B B B =...
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lectures29-31 - Kinetics of Phase Transformations...

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