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Unformatted text preview: ME 382 Lecture 15 10/ii/06 1 I NTRODUCTION TO K INETICS (2) Effects of phase boundaries Nucleation rate of phase changes • Consider phase change α → β at a temperature T 1 < T e (super-cooling) • Decrease in Gibb’s free energy Δ G αβ = G α- G b J / unit volume transformed material • Increase in G caused by the energy of the interface between the two phases γ αβ J / unit area of interface between the two phases • Total change in G by transforming a volume V of α to β forming an interfacial area A : ! G total = " V # ! G $% + A & $% • Consider spherical particles of β of radius r forming in α V = 4 3 ! r 3 A = 4 ! r 2 ∴ " G total = # 4 3 $ r 3 " G %& + 4 $ r 2 ’ %& • Critical nucleus size is r o • If r < r o : growing nucleus increases energy • If r > r o : growing nucleus decreases energy • r o given by finding r where d Δ G total / d r = 0 dG total dr = ! 4 " r 2 # G $% + 8 " r & $% ∴ dG total dr = when r o = 2 ! "# / $ G "# ME 382 Lecture 15 10/ii/06 2 • But ! G "# = ! S "# T e $ T 1 ( ) = ! H "# T e T e $ T 1 ( ) (see previous lecture) ∴ r o = 2 " #$ T e % H #$ T e & T 1 ( ) • Larger degrees of undercooling ⇒ smaller critical radii • Smaller clusters are more likely to form randomly than larger clusters ∴ More likely to find critical sized nucleii with more undercooling...
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- Fall '08
- Gtotal, phase boundaries Nucleation, nucleation sites