Question

[5 Points] In the solidification of a pure metal, what are the two energies involved in the transformation? Write the equation for the total free-energy change involved in the transformation of liquid to produce a strain-free solid nucleus by homogeneous nucleation 1. 2. [5 Points] In the solidification of a metal, what is the difference between an embryo and a nucleus? 3. 5 Points] Distinguish between homogeneous and heterogeneous nucleation for the solidification of a pu re metal. 4. [5 Points] Calculate the size (radius) of the critically sized nucleus for pure platinum when homogeneous nucleation takes place. 5. [5 Points] Calculate the number of atoms in a critically sized nucleus for the homogeneous nucleation of pure platin um 6. [15 Points] Calculate the radius of the largest interstitial void in the BCC a iron lattice. The atomic radius of the iron atom in this lattice is 0.124 nm, and the largest interstitial voids occur at the (X, , 0); (, , 0); (%, %, O); (%, %, O), etc., -type positions. 7. [20 Points] Calculate the energy for vacancy formation in silver, given that the equilibrium number of vacancies at 800 °C is 3.6 x 1023 m3. The atomic weight and density (at 800°c) for silver are, respectively, 107.9 g/mol and 9.5 g/cm [20 Points] The expression for the total free energy change for nucleation for a spherical shaped 8. nucleus is: AG= raG, + 4mry For the case of a cubic nucleus of edge length a (instead of a sphere of radius r), write an expression for the total free energy change for nucleation and then solve for both the critical cube edge length, a and also AG* (Hint: differentiate this expression with respect to a). 9. [20 Points] It is known that the kinetics of recrystallization for some alloy obey the Avrami equation, and that the value of n in the exponential is 5.0. If, at some temperature, the fraction recrystallized is 0.30 after 100 min, determine the rate of recrystallization at this temperature

Material and Science Engineering (chemistry and physics II)

Answer 1

1)

In the solidification of a pure metal, the two energies involved in the transformation are:- (a) The volume free-energy released by the liquid to solid transformation. (b) The surface free-energy required to form the new solid surfaces of the solidified particles. (a) The volume free-energy released by the liquid to solid transformation:- When a liquid is allowed to solidify it must release some amount of energy while this liquid to solid transformation takes place. In case of solidification of pure metal such as lead is cooled below its equilibrium freezing temperature, the energy needed for liquid to solid transformation is difference of volume free-energy (AG) of liquid and that of the solid. If AG is the change in free energy between the solid and liquid per unit volume of the metal, then the free energy change for a spherical nucleus of the radius ris ar AG, Where ar' is the volume of spherical nucleus. The change in volume free-energy versus radius of nucleus is shown by an lower negative curve below which is a negative quantity since energy is released by the liquid to solid transformation.

Retarding energy Total free-energy change Free-energy change (AG) Radius of particle Driving energy

(b) The surface energy required to form the new solid surfaces of the solidified particles:- it is the retarding energy required to form the surface of particles like embryos and nuclei. If we are talking about spherical particles, the retarding energy (AG) needed to create a surface for these spherical particles are equal to y times the area of the surface of the sphere or AG = 4ary, where 4xris surface area of a sphere. This retarding energy (AG) for the formation of solid particles is shown by an upward positive curve in the figure. Total free-energy equation:- The associated total energy during the formation of an embryo or nucleus as discussed above, which is the sum of the volume free-energy and surface free-energy is shown in the above figure as middle curve. The total free-energy for the formation of a spherical nucleus of radius rformed in a freezing pure metal is given as AG = Ar AG,+ 4ary Where AG, = Total free-energy change r = radius of embryo or nucleus AG, = Volume free-energy y = Specific surface free-energy

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