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lastsetoflectures - Simultaneous Ordering and Phase...

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Unformatted text preview: Simultaneous Ordering and Phase Separation Turbine Blade microstructure Atomic structure Ni Al ' + ' Microscopic diffusion equation simulation of phase separation A B G A B n G g G o G p G Nucleation and Growth Continuous transformations Nucleation and Growth vs. Continuous Transformations G B X G G G 2 2 > B X G 2 2 < B X G spinodal meta- stable meta- stable stable stable 2 2 > B X G spinodal or continuous n fluctuatio small 2 2 < G X G B < D ( ) x c B x < D ( ) x c B x c c up-hill diffusion homogeneous phase two phase c o solution s homogeneou n fluctuatio large n fluctuatio small growth and nucleation n fluctuatio large n fluctuatio small 2 2 > G G G G X G B ( ) x c B x ( ) x c B x c c Nucleation and growth c o c c > D Theory of Nucleation 1. Theory of nucleation for the formation of liquid droplets in supersaturated vapors r vapor liquid droplet Assumption : the small droplet has the same surface and thermodynamic properties as the bulk product phase The free energy change accompanying the formation of a new particle is given by A g V G + = V volume of particle, A surface area, g chemical free energy change per unit volume of the product formed, is the specific interfacial energy For a spherical particle of radius r 2 3 4 3 4 r g r G + = For a given phase transformation, < g > and G r * G * r ( ) * 2 * 8 4 * r g r r G r r + = = = g r = 2 * ( ) 2 3 * 3 16 g G = r *- critical nucleus size G *- critical free energy of formation of a critical nucleus The free energy change, g , can be approximated by ( with the assumption that heat capacity difference between parent and product phases is independent of temperature ) ( ) o o o T T h T T T h g = h enthalpy change (or heat of transformation) per unit volume of the product formed, T o is the equilibrium temperature at which g = 0, and T is the degree of undercooling. Therefore, the critical free energy of formation can be written as ( ) ( ) 2 2 2 3 * 3 16 T h T G o = If the total number of particles per unit volume of the matrix is N T , the number of embryos (clusters) for a given size can be approximated by ( ) T k G N N B T = exp where G is the free energy change for the formation of an embryo. Therefore, the number of critical nuclei is given by ( ) T k G N N B T * * exp = In classical theory, the process of nucleation is defined by the addition of one atom to a critical-sized particle....
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lastsetoflectures - Simultaneous Ordering and Phase...

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