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Askeland Science and Engineering 7e ISM Chapter 12

Askeland Science and Engineering 7e ISM Chapter 12

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Chapter 12: Dispersion Strengthening by Phase Transformations and Heat Treatment 12-1 (a) Determine the critical nucleus size r* for homogeneous nucleation for precipitation of phase β in a matrix of phase α. Hint: The critical nucleus size occurs at the maximum in the expression for ΔG(r) in Equation 12–1. (b) Plot the total free energy change ΔG as a function of the radius of the precipitate. (c) Comment on the value of r* for homogeneous nucleation for solid-state precipitation when compared to the liquid to solid transformation. Solution: (a) From Equation 12–1, the total change in free energy for nucleation of a spherical solid precipitate ΔG is 4 4 ∆G = π r 2 ∆Gv (α → β ) + 4π r 2σ αβ + π r 3ε , 3 3 where r is the radius of spherical solid, ΔGv(α→β) is the free energy change per unit volume for precipitation of phase β from phase α, σαβ is the surface energy per unit area of the α–β interface, and is the strain energy per unit volume required for a precipitate to fit into the surrounding matrix. Rearranging, 4 ∆G = π r 3 [ ∆Gv (α → β ) + ε ] + 4π r 2σ αβ . 3 The critical radius for nucleation r* is the value at which ΔG is a maximum. At this point, the slope is zero. Thus, ∂∆G = 4π r 2 [ ∆Gv (α → β ) + ε ] + 8π rσ αβ , ∂r and ∂∆G ∂r = 4π (r*) 2 [∆Gv (α →β ) + ε ] + 8π r *σ αβ = 0 . r =r* Solving for r*, r* = − 2σ αβ ∆Gv (α → β ) + ε . (b) A graph of total fre ...
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