Fraction transformed in isothermal process – Avrami analysis Consider    transformation How do we determine the volume (or area) fraction transformed?  How do you deal with the overlap? Mathematical device : extended volume fraction Xex  volume fraction transformed disregarding overlap.

The actual volume fraction grows in a relative amount to the unconsumed fraction, at the same rate the extended volume fraction does.: or Unconsumed fraction Integrate Avrami equation Expand : dilute overlap of two overlap of three

Application to nucleation & growth : ( Johnson - Mehl) Case (1) constant number of heterogeneous nuclei present from the beginning. concentration: N growth rate of crystals : v x t Plot of ln t vs ln[-ln(1-x)] should have slope of 3.

Plot of ln t vs ln[-ln(1-x)]  slope of 4 Case (2) Assume a constant nucleation rate I, # of nuclei formed between t’ and t’ + dt’ ; concentration, N = I dt’ and at some later time ( t > t’ ) the “radius” of transformed phase is v (t – t’) so Plot of ln t vs ln[-ln(1-x)]  slope of 4 These plots are called Johnson- Mehl –Arami plots (JMA plots) c

Case study : Devitrification of Au65Cu12Si9Ge14 glass C. Thompson et. al., Acta Met., 31, 1883 (1983) Calorimetry results Fraction transformed power DSC isothermals Time (min) 20 40 60 80 100 329K 328K 327K 326K 325K 324K Time 1/2 1 X 329K 328K 327K 326K 325K 324K

JMA plot (327K) ln [-ln(1-x)] ln [-ln(1-x)] ln (1-t) ln (t) slope = 4 ln (t) must be introduced N = Iss(t -) Slope = 4.0

Time-Temperature-Transformation Curves TTT curves” are a way of plotting transformation kinetics on a plot of temperature vs. time. A point on a curve tells the extent of transformation in a sample that is transformed isothermally at that temperature. A TTT diagram shows curves that connect points of equal volume fraction transformed. l+ β l T l+ α α β α +β A xB → B

Time-Temperature-Transformation Curves Curves on a TTT diagram have a characteristic “C” shape that is easily understood using phase transformations concepts. The temperature at which the transformation kinetics are fastest is called the “nose” (•) of the TTT diagram A TTT diagram shows curves that connect points of equal volume fraction transformed.

Construction of TTT diagrams from Avrami Curves x log t 100% transformed decreasing T T1 50% transformed T4 50% transformed T4 trans start: 0 transformed T1 Temp log time

Construction of TTT diagrams from Avrami Curves Fe-C phase diagram

Fe-C phase diagram: Perlite

Fe-C TTT diagram example