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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: orderdisorder, 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 . CahnHilliard Diffusion Equation and Spinodal Decomposition • DiffuseInterface Theory • Equilibrium Composition Profile • Interfacial Free Energy of a Flat Interface • Interfacial Width • UpHill Diffusion • CahnHilliard Diffusion • Kinetics of Spinodal Decomposition o X ' α X ' ' α X Phase Separation T X α " ' α α + MonteCarlo Simulation of phase separation Microscopic diffusion equation simulation of phase separation Examples : liquids, glasses, polymer blends, solids such as AlZn, TiO 2SnO 2 , AlCu, CoOMgO, etc. ( ) ( ) ( ) B B B v X f X f dx A dx X f A F β β α α ω ω + = = ∫ ∫ ∞ + ∞ − ∞ + ∞ − v It’s only a function of volume fraction Free Energy of a TwoPhase Mixture ( ) x c B x β c α 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 α − α ω phase of fraction volume β ω β − e unit volum per phase of energy free α f − α e unit volum per phase of energy free β β − f n compositio − B X area sectional cross − A The DiffuseInterface Theory ( ) ( ) [ ] ∫ ∇ + = v 2 v v 2 1 dV X X f V F B B κ In general, the local free energy depends not only the local composition but also the composition of the immediate environment. In the diffuseinterface description, the local free energy is expressed as a function of the local composition and the local composition gradient, respectively, i.e. where κ 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 κ ( ) 2 2 x X X X f X F B B B B ∂ ∂ − ∂ ∂ = κ δ δ ( ) ( ) ∫ = dx dx dy x y x I Y , , ( ) ( ) dx dy I dx d y I x y Y ∂ ∂ − ∂ ∂ = δ δ 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 ∂ ∂ − ∂ ∂ = κ δ δ...
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This note was uploaded on 01/19/2010 for the course MAT SCI 503 taught by Professor Chen during the Spring '02 term at Penn State.
 Spring '02
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