Lecture Problem 11'2

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Lecture Problem 11.2

• Corina Tom
• 4-13-07
• E106 - Section 2

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Problem Statement

• Rewrite the expression for the total free energy
  change for nucleation (Eqn. 11.1) for the case of
  a cubic nucleus of edge length a (instead of a
  sphere with radius r). Now differentiate this
  expression with respect to a and solve for both
  the critical cube edge length, a, and also ?G.
• Is ?G greater for a cube or a sphere? Why?

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Nucleation

• First stage in a phase transformation
• Very small particles (nuclei) of new phase appear
• Homogeneous - nuclei of new phase form uniformly
  throughout parent phase
• Heterogeneous - nuclei of new phase form
  preferentially at structural inhomogeneities

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?G

• G Free Energy (Gibbs Free Energy)
• When ?G lt 0, a spontaneous transformation will
  occur!
• ?Gv volume free energy
• Negative if TltTm
• Magnitude of contribution volume ?Gv
• ? surface free energy
• Positive
• Magnitude of contribution surface area?

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Spherical Nucleus

• Radius r

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Figure 11.2 from Callister
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• Differentiate to find maximum
• Critical radius r
• Critical free energy ?G

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Cubic Nucleus

• Edge length a
• Volume a3
• Surface area 6a2

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• Differentiate to find maximum

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Is ?G greater for a cube or a sphere?

• Easy way

Cube has greater ?G
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• More conceptual way
• Volume term gt GO
• Surface area term gt DONT GO
• For fixed volume (1 unit3), check SA

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The Big Picture

• Heat Treatment gt Phase Transformations
• Phase Transformations gt Microstructures
• Microstructures gt Material Properties
• So, we can design a heat treatment that results
  in a particular set of properties

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Example Application

• Iron-Carbon alloy
• Eutectoid composition (0.76 wt C)
• Tensile strength 700 MPa - 2000 MPa

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Questions?
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