Northeastern University Kinetics of Phase Transformations: Nucleation & Growth Radhika Barua Department of Chemical Engineering Northeastern University Boston, MA USA Northeastern University Thermodynamics of Phase Transformation For phase transformations (constant T & P) relative stability of the system is defined by its Gibb’s free energy (G). • Gibb’s free energy of a system: G • G=H-TS ΔGa • Criterion for stability: dG=0 dG=0 • dG=0 ΔG • Criterion for phase transformation: • ΔG= GA-GB < 0 B Activated A State But …… How fast does the phase transformation occur ? 2 Northeastern University Kinetics of Phase Transformation Phase transformations in metals/alloys occur by nucleation and growth. • Nucleation: New phase (β) appears at certain sites within the metastable parent (α) phase. • Homogeneous Nucleation: Occurs spontaneously & randomly without preferential nucleation site. • Heterogeneous Nucleation: Occurs at preferential sites such as grain boundaries, dislocations or impurities. • Growth: Nuclei grows into the surrounding matrix. SOLID LIQUID Example: Solidification , L S (Transformations between crystallographic & non-crystallographic states) 3 Northeastern University Driving force for solidification Example: Solidification , L S • At a temperature T: L L L S S S Driving Force for • G = H - TS ; G = H - TS solidification • ΔG = GL – GS = ΔH – TΔS ΔG • At the equilibrium melting point (Tm): • ΔG = ΔH – TmΔS = 0 GS • ΔH = L (Latent heat of fusion) Free energy (G) • For small undercoolings (ΔT): ΔT GL • ΔG ≈ L ΔT Tm T TM Temperature Decrease in free energy (ΔG) provides the driving force for solidification Northeastern University Homogeneous Nucleation • Difference in free energy: • ΔGhom = G1 – G2 = V(Gs – GL) + AγSL • For a spherical particle: • ΔGhom = G1 – G2 SOLID LIQUID LIQUID Volume Interfacial G2 G1 free energy energy • Note the following: • Volume free energy increases as –r3 • Interfacial free energy increases as r2 Northeastern University ΔGhom for a given undercooling (ΔT) Interfacial energy α r2 r* ΔG*hom ΔT GL Volume free energy α r3 GS’ GS r=r* ΔG=2γ/r * r = ∞ ΔT Note : Both r* and ΔG* depend on undercooling (ΔT). 6 T TM Northeastern University Critical Undercooling for Nucleation Assumptions: • Liquid with nuclei is an ideal solution of various size clusters. • Each size cluster contains i atoms or molecules. Homogeneous nucleation occurs only when liquid is undercooled by TN Critical undercooling for nucleation 7 Northeastern University Rate of Homogeneous Nucleation 3 • For a given undercooling: clusters/m Note: C0 , Atoms per unit volume in the liquid. C*, Number of atoms that have reached critical size. • Addition of one more atom, converts the clusters to a stable nuclei. -3 -1 • If this happens with a frequency of f0: Nuclei / m S -3 -1 Nuclei / m S N ΔT ΔT N No nuclei is formed until ΔT is reached !! N 8 Northeastern University Heterogeneous Nucleation In practice, homogeneous nucleation is rarely observed. Sources of nucleation sites: • Dislocations • Grain boundaries • Dust particles • Secondary phase particles • Mould walls & cracks ΔGhet = V(Gs – GL) + ASLγSL + ASMγSM - ASMγML = where, S(θ) ≤ 1 is a function of the wetting angle 9 Northeastern University ΔGhet for a given undercooling (ΔT) ΔG * ΔG hom * ΔG het r r* ΔGhet Note: ΔGhom • r* depends only on ΔT. • ΔG*het depends of S(θ) & ΔT • ΔG*het < ΔG*hom 10 Northeastern University Variation of ΔG* & nucleation rates with ΔT Smaller undercooling is required for heterogeneous nucleation -3 -1 Nuclei / m S where, f1 is the frequency factor C1 is the # of atoms in contact with the heterogeneous nucleation sites. 11 Northeastern University Avrami Model for Growth Assumptions: • Nucleation occurs randomly and homogeneously • Growth rate does not depend on the extent of transformation • Growth occurs at the same rate in all directions Nuclei Parent phase New secondary phase Ref: www.wikipedia.com 12 Northeastern University Avrami Model: Derivation NOTE: , where G’ & N’ are the growth and nucleation rates n = 4 ……… when growth is 3-D & N’ is constant n = 3 ……… when growth is 3-D & nuclei are preformed n = 1,2 …… when growth is restricted in 1-D (surface) or 2 D (edge) 2-D growth along a stepped interface 13 Northeastern University First order magnetostructural transitions First order magnetostrucutral transitions share common features with solidification. Example: Bulk Fe1-xRhx (0.485 < x < 0.55) AFM (Levitin, Soviet Physics JETP, 1966) phase Phase transition features: FM phase • Thermal hysteresis • Tt = f(H,P) • ~ 1% volume expansion (Kouvel and Hartelius, J. Appl. Phys ,1962) 14 Northeastern University Thermal hysteresis Example: Hypothesized FeRh nanoparticles in Cu matrix. Onset of Phase #1 Complete transformation of Phase #2 Complete transformation of Phase #1 T~130 Nucleation of Phase #2 K Type II AFM FM Phase #1: AFM ??? PHASE #2: FM ??? 15 Northeastern University Extension of Avrami Equation Minor thermal hysteresis loops during heating & cooling Temperature dependance of area of minor loops Reference: Manekar and Roy, J. Phys.: Condens. Matter 20 (2008 .