# hw 7 - [010 ME311 MATERIALS ENGINEERING[110 2011 SPRING...

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[010] φ [110] θ ME311 MATERIALS ENGINEERING SPRING 2011 HOMEWORK #7 DUE DATE: March 25, before class PROBLEM 1: Estimate the number of spherical clusters in 1mm 3 of copper at its melting point that contain a) 10 atoms and b) 60 atoms. The atomic volume of liquid copper is 1.6x10 -29 m 3 , γ SL is 0.177 J/m 2 , k =1.38x10 –23 J/K and T m is 1356 K. Hint: Consider that at the melting temperature, the free energies per unit volume of the liquid and the solid are the same. SOLUTION At the equilibrium melting temperature T m we have In addition, at the equilibrium melting temperature , so that For a cluster containing n c atoms with an atomic volume we have . Therefore the expression for G r becomes Substituting = 1.6 × 10 -29 m 3 and γ = 0.177 Jm -2 gives . For 1 mm 3 , n 0 = 6.25 × 10 19 atoms . Therefore when n c = 10 atoms, n r = 9 × 10 13 clusters mm -3 ; and when n c = 60 atoms, n r = 3 clusters mm -3 . PROBLEM 2: Copper (FCC crystal structure) is normally alloyed with additional elements in order to increase its strength. For an alloy with a composition of Cu (80 wt%) and Ag (20 wt%) (check the Cu-Ag phase diagram), when the first solid nucleates from the liquid,

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