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Chapter 10: Phase Transformations

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Chapter 10: Phase Transformations Chapter 10: Phase Transformations ISSUES TO ADDRESS... • Transforming one phase into another takes time. Fe Fe C Eutectoid 3 g transformation (cementite) (Austenite) + a C FCC (ferrite) (BCC) • How does the rate of transformation depend on time and T? • How can we slow down the transformation so that we can engineering non-equilibrium structures? • Are the mechanical properties of non-equilibrium structures better? Chapter 10 - 1 Phase transformation • Takes time (transformation rates: kinetics). • Involves movement/rearrangement of atoms. • Usually involves changes in microstructure 1. “Simple” diffusion-dependent transformation: no change in number of compositions of phases present (e.g. solidification of pure elemental metals, allotropic transformation, recrystallization, grain growth). 2. Diffusion-dependent transformation: transformation with alteration in phase composition and, often, with changes in number of phases present (e.g. eutectoid reaction). 3. Diffusionless transformation: e.g. rapid T quenching to “trap” metastable phases. Chapter 10 - 2 º TR = recrystallization temperature TR Adapted from Fig. 7.22, Callister 7e. The influence of annealing T on the tensile strength and ductility of a brass alloy. º Chapter 10 - 3 Iron-Carbon Phase Diagram Eutectoid cooling: cool g (0.76 wt% C) a (0.022 wt% C) + Fe3C (6.7 wt% C) T(°C) heat 1600 d 1400 L g g+L 1200 A L+Fe C (austenite) 1148°C 3 R S 1000 g g g+Fe C g g 3 800a B 727°C = Teutectoid R S (cementite) C 3 600 a+Fe C 3 Fe 400 0 1 2 3 4 5 6 6.7 0.76 4.30 (Fe) Co, wt% C Fe3C (cementite-hard) a (ferrite-soft) eutectoid Adapted from Fig. 9.24,Callister 7e. C Chapter 10 - 4 Phase Transformations Nucleation – nuclei (seeds) act as template to grow crystals – for nucleus to form rate of addition of atoms to nucleus must be faster than rate of loss – once nucleated, grow until reach equilibrium Driving force to nucleate increases as we increase T – supercooling (eutectic, eutectoid) – superheating (peritectic) Small supercooling few nuclei - large crystals Large supercooling rapid nucleation - many nuclei, small crystals Chapter 10 - 5 Solidification: Nucleation Processes • Homogeneous nucleation – nuclei form in the bulk of liquid metal – requires supercooling (typically 80-300°C max) • Heterogeneous nucleation – much easier since stable “nucleus” is already present • Could be wall of mold or impurities in the liquid phase – allows solidification with only 0.1-10ºC supercooling Chapter 10 - 6 Phase 1 (e.g. liquid) Nucleation of 2nd phase Growth Homogeneous Nucleation Thermodynamic parameters: • Free energy G (or Gibbs free energy) • Enthalpy H: internal energy of the system and the product of its volume multiplied by the pressure • Entropy S: randomness or disorder of the atoms or molecules ΔG is important---a phase transformation will occur spontaneously only when ΔG has a negative value. ΔG = ΔH - TΔS Chapter 10 - 7 Homogeneous Nucleation Chapter 10 - 8 Homogeneous Nucleation & Energy Effects Surface Free Energy- destabilizes the nuclei (it takes energy to make an interface) 2 GS 4r g g = surface energy Activation Free Energy GT = Total Free Energy = GS + GV Volume (Bulk) Free Energy – stabilizes the nuclei (releases energy) 4 G r 3G V 3 volume free energy G unit volume r* = critical nucleus: nuclei < r* shrink; nuclei>r* grow (to reduce energy) Adapted from Fig.10.2(b), Callister 7e. Chapter 10 - 9 Kinetics of solid state reactions Critical nucleus size (rc) and the activation energy (G*) 4 G = r3G 4r 2g 3 v Take the derivative and set equal to zero to find max. d(G) 4 (G )(3r 2 ) 4g (2r) 0 dr 3 v 2g rc Gv Substitution in to overall G equation 3 * 16g G 2 3(Gv ) Chapter 10 -10 The volume free energy change is the driving force for the solidification transformation Kinetics of solid state reactions In terms of latent heat of fusion Hf (i.e. energy release upon solidification): H (T T) f m Tells us how Gv changes with temperature Gv Equilibrium Tm solidification temperature With this definition, we then have: T1 > T2 2g T m rc H f Tm T 2 16g 3 T G* m 2 3H f Tm T As T decreases both rc * * G and G* become smaller Number of stable nuclei: n exp kT Chapter 10 - 11 LIQUID INSTABILITY at LOWER TEMPERTURES Solidification r* = critical radius 2gT r* m g = surface free energy H T S Tm = melting temperature HS = latent heat of solidification T = Tm - T = supercooling Note: HS = strong function of T g = weak function of T r* decreases as T increases For typical T r* ca. 100Å Chapter 10 -12 Kinetics of solid state reactions We also need to consider diffusion: • Faster diffusion leads to more collisions between atoms. • More collisions means higher probability of atoms sticking to each other. Recall diffusion Qd D Do exp kT Then, the frequency of atoms sticking together is directly related to diffusion: Qd Frequency of attachment: vd exp kT Chapter 10 -13 Kinetics of solid state reactions Combining liquid instability and diffusion effects together: Rate of Nucleation (units: nuclei per unit volume per second) dN G* Qd Kn*vd K'exp exp Contribution from Contribution from dt kT kT liquid instability diffusion Net rate Qd vd exp rate Nucleation kT G* Liquid instability Diffusion n* exp Tm T kT Chapter 10 -14 Example problem: critical radius and activation energy for nucleation A) If pure liquid gold is cooled to 230oC below its melting point, calculate the critical radius and the activation energy. Values for the latent heat of fusion and surface free energy are -1.16x109J/m3 and 0.132J/m2, respectively. 2g T m rc H f Tm T 2 16g 3 1.32nm T 9.64x10-19J G* m B) Calculate3H 2 Tthe numberT of atoms per nucleus of this critical size. Au is FCC withf am = 0.413nm. 4 r *3 critical nucleus volume # unit cells/particle 3 137 unit cell volume a3 4 atoms/unit cell……548 atoms/critical nucleus Chapter 10 -15 Rate of Phase Transformations Kinetics – time dependence of nucleation, growth and transformation rates. • Hold temperature constant & measure transformation vs. time How is transformation measured? • microscopic examination X-ray diffraction – have to do many samples electrical conductivity – follow one sample sound waves – one sample Chapter 10 -16 Rate of Phase Transformation y All out of material - done Fixed T 0.5 maximum rate reached – now amount unconverted decreases so rate slows rate increases as surface area increases t0.5 & nuclei grow Fraction transformed, transformed, Fraction log t Adapted from Fig. 10.10, Callister 7e. Avrami rate equation => y = 1- exp (-ktn) fraction time transformed – k & n fit for specific sample By convention r = 1 / t0.5 Chapter 10 -17 Rate of Phase Transformations Cu Adapted from Fig. 10.11, Callister 7e. (Fig. 10.11 adapted 135C 119C 113C 102C 88C 43C from B.F. Decker and D. Harker, "Recrystallization in Rolled Copper", Trans AIME, 188, 1950, p. 888.) 1 10 102 104 • In general, rate increases as T -Q/RT r = 1/t0.5 = A e – R = gas constant Arrhenius – T = temperature (K) expression – A = preexponential factor – Q = activation energy r equilibrium not possible! • often small: Chapter 10 -18 Eutectoid cooling: cool g (0.76 wt% C) a (0.022 wt% C) heat + Fe3C (6.7 wt% C) Chapter 10 -19 Eutectoid Transformation Rate • Growth of pearlite from austenite: Diffusive flow of C needed Austenite (g) cementite (Fe3C) grain a Ferrite (a) a boundary a g g g a g Adapted from a pearlite a a Fig. 9.15, growth Callister 7e. a direction a • Recrystallization 100 600°C rate increases (T larger) 650°C with T. 50 Adapted from 675°C Fig. 10.12, (T smaller) Callister 7e. (% pearlite) y 0 Course pearlite formed at higher T - softer Fine pearlite formed at low T - harder Chapter 10 -20 Nucleation and Growth • Reaction rate is a result of nucleation and growth of crystals. 100 Nucleation rate increases with T % Pearlite Growth regime Growth rate increases with T 50 Nucleation regime Adapted from 0 t0.5 log (time) Fig. 10.10, Callister 7e. • Examples: pearlite g colony g g T just below TE T moderately below TE T way below TE Nucleation rate low Nucleation rate med . Nucleation rate high Growth rate high Growth rate med. Growth rate low Chapter 10 -21 Transformations & Undercooling • Eutectoid transf. (Fe-C System): g a + Fe3C • Can make it occur at: 0.76 wt% C 6.7 wt% C ...727ºC (cool it slowly) 0.022 wt% C ...below 727ºC (“undercool” it!) T(°C) 1600 Adapted from Fig. d 9.24,Callister 7e. (Fig. 9.24 1400 L adapted from Binary Alloy Phase Diagrams, 2nd ed., g g+L Vol. 1, T.B. Massalski (Ed.- 1200 1148°C L+Fe C in-Chief), ASM International, (austenite) 3 Materials Park, OH, 1990.) 1000 +Fe C a Eutectoid: g 3 ferrite 800 Equil. Cooling: Ttransf. = 727ºC 727°C T (cementite) C a +Fe3C 3 600 Undercooling by T < 727C transf. Fe 400 0.76 0 0.022 1 2 3 4 5 6 6.7 (Fe) Co, wt%C Chapter 10 -22 Isothermal Transformation Diagrams T-T-T plots • Fe-C system, Co = 0.76 wt% C • Transformation at T = 675°C. T: hold constant 100 T = 675°C , y 50 0 % transformed 1 102 104 time (s) Only for Fe-C alloy of T(°C) Austenite (stable) eutectoid composition T (727C) 700 Austenite E (unstable) Pearlite 600 Adapted from Fig. 10.13,Callister 7e. isothermal transformation at 675°C (Fig. 10.13 adapted from H. Boyer (Ed.) Atlas of Isothermal Transformation and 500 Cooling Transformation Diagrams, American Society for Metals, 1977, p.
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