1MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHeat Treatment (time and temperature) ⇒ MicrostructureChapter Outline: Phase Transformations¾ Kinetics of phase transformations¾ Homogeneous and heterogeneous nucleation¾ Growth, rate of the phase transformation¾ Metastable and equilibrium states¾ Phase transformations in Fe-C alloys¾ Isothermal Transformation DiagramsNot tested: in 10.5 Bainite, Spheroidite, Martensite… (from p. 360)10.6 Continuous Cooling Transformation Diagrams10.7-10.9 Mechanical behavior of Fe-C alloys, Tempered Martensite2MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsPhase transformations (change of the microstructure) can be divided into three categories:Phase transformations. Kinetics.¾ Diffusion-dependent with no change in phase composition or number of phases present (e.g. melting, solidification of pure metal, allotropic transformations, recrystallization, etc.)¾ Diffusion-dependent with changes in phase compositions and/or number of phases (e.g. eutectic or eutectoid transformations)¾ Diffusionless phase transformation - by cooperative small displacements of all atoms in structure, e.g. martensitic transformation (discussed in this chapter but not tested)Phase transformations do not occur instantaneously. Diffusion-dependent phase transformations can be rather slow and the final structure often depend on the rate of cooling/heating. We need to consider the time dependence or kinetics of the phase transformations.3MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsPhase transformations involve change in structure and (for multi-phase systems) composition ⇒ rearrangement and redistribution of atoms via diffusion is required.The process of phase transformation involves:Kinetics of phase transformations¾ Nucleation of the new phase(s) - formation of stable small particles (nuclei) of the new phase(s). Nuclei are often formed at grain boundaries and other defects.¾ Growth of the new phase(s) at the expense of the original phase(s).S-shape curve: percent of material transformed vs. the logarithm of time.4MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsNucleationNucleation can be Heterogeneous – the new phase appears on the walls of the container, at impurity particles, etc.Homogeneous – solid nuclei spontaneously appear within the undercooled phase.Let’s consider solidification of a liquid phase undercooled below the melting temperature as a simple example of a phase transformation.solidsolidliquid liquidhomogeneousnucleationheterogeneousnucleationsupercooledliquid5MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsGibbs free energy in analysis of phase transitionsIt is convenient to analyze phase transformations occurring under conditions of constant temperature (T) and pressure (P) by using Gibbs free energy (G).G = H – TS, where H is the enthalpy and S is the entropyH = U + PV, where U is the internal energyThe conclusion of the thermodynamic analysis (take MSE 3050 to learn about the details) is that the equilibrium under conditions of T = const, and P = const corresponds to the minimum of G and a phase transformation occurs spontaneously only when G decreases in the course of the transformationGibbs free energy (G = H – TS): Equilibrium is trade-off between minimization of enthalpy and maximization of entropyA change to a lower enthalpy state (ΔH < 0) usually decreases the randomness (ΔS < 0):6MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHomogeneous nucleationsolidliquidIs the transition from undercooled liquid to a solid spherical particle in the liquid a spontaneous one?That is, is the Gibbs free energy decreases?supercooledliquidThe formation of a solid nucleus leads to a Gibbs free energy change of ΔG = G2-G1= -VS (GvL–GvS) + ASLγSLnegative below Tmalways positive12VSis the volume of the solid sphereASLis the solid/liquid interfacial areaγSLis the solid/liquid interfacial energyΔGv= GvL–GvSis the volume free energy differenceat T < Tm, GvS< GvL– solid is the equilibrium phase7MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHomogeneous nucleationΔG = G2-G1= -VS Δ Gv+ ASLγSLFor a spherical nucleus with radius r:3Sr π34V =SL2v3γr 4πΔGr π34-ΔG +=2SLr 4πA =rΔG*ΔG*rΔGinterfacial energy ~ r2volume energy ~ r3For nucleus with a radius r > r*, the Gibbs free energy will decrease if the nucleus grows. r* is the critical nucleus size.8MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHomogeneous nucleationAt r = r*0r γ 8πΔGr -4 πdrG dΔSLv2=+=vSL*ΔG γ2r =()()2v3SL*ΔG3γ 16πΔG =mmvTΔTΔHΔG =ΔT1ΔHT γ2rmmSL*⎟⎟⎠⎞⎜⎜⎝⎛=()()()22m2m3SL*ΔT1ΔH3Tγ 16ΔG⎟⎟⎠⎞⎜⎜⎝⎛π=Both r* and G* decrease with increasing undercooling The difference between the Gibbs free energy of liquid and solid (also called “driving force” for the phase transformation) is proportional to the undercooling below the melting temperature, ΔT = Tm–T:where Hmis the latent heat of melting (or fusion) Therefore:9MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHomogeneous nucleationΔT1ΔHT γ2rmmSL*⎟⎟⎠⎞⎜⎜⎝⎛=()()()22m2m3SL*ΔT1ΔH3Tγ 16ΔG⎟⎟⎠⎞⎜⎜⎝⎛π=Both r* and G* decrease with increasing undercooling rΔG*ΔG1*1rΔG*2r*ΔG2m12TTT<<10MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsHomogeneous nucleationThere is an energy barrier of ΔG* for formation of a solid nucleus of critical size r*. The probability of energy fluctuation of size ΔG* is given by the Arrheniusequation and the rate of homogeneous nucleation is⎟⎟⎠⎞⎜⎜⎝⎛−νkTΔGexpN*d~&nuclei per m3per swhere νdis the frequency with which atoms from liquid attach to the solid nucleus. The rearrangement of atoms needed for joining the solid nucleus follow the same temperature dependence as the diffusion coefficient:⎟⎠⎞⎜⎝⎛−νkTexpddQ~⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛−kTΔGexpkTQexpN*d~&Therefore:11MSE 2090: Introduction to Materials Science Chapter 10, Phase TransformationsRate of homogeneous nucleation⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛−kTΔGexpkTQexpN*d~&⎟⎟⎠⎞⎜⎜⎝⎛−kTΔGexp*⎟⎠⎞⎜⎝⎛−kTQexpdN&mTTemperatureΔG* is too high - nucleation is