Kinetics of Phase Transformations: Nucleation & Growth

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1 Kinetics of Phase Transformations: Nucleation & Growth Radhika Barua Department of Chemical Engineering Northeastern University Boston, MA USA

2 Thermodynamics of Phase Transformation Northeastern University 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=H-TS Criterion for stability: dg=0 Criterion for phase transformation: ΔG G dg=0 ΔG a dg=0 ΔG= G A -G B < 0 B Activated State A But How fast does the phase transformation occur? 2

3 Kinetics of Phase Transformation Northeastern University 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 Example: Solidification, L S LIQUID (Transformations between crystallographic & non-crystallographic states) 3

4 Driving force for solidification Example: Solidification, L S At a temperature T: G L = H L - TS L ; G S = H S - TS S ΔG = G L G S = ΔH TΔS At the equilibrium melting point (T m ): ΔG = ΔH T m ΔS = 0 ΔH = L (Latent heat of fusion) Free energy (G) ΔG Driving Force for solidification G S For small undercoolings (ΔT): ΔG L ΔT ΔT G L T m T T M Temperature Decrease in free energy (ΔG) provides the driving force for solidification

5 Homogeneous Nucleation Difference in free energy: ΔG hom = G 1 G 2 = V(G s G L ) + Aγ SL For a spherical particle: ΔG hom = G 1 G 2 SOLID LIQUID LIQUID Volume free energy Interfacial energy G 1 G 2 Note the following: Volume free energy increases as r 3 Interfacial free energy increases as r 2

6 ΔG hom for a given undercooling (ΔT) Northeastern University r * Interfacial energy α r 2 ΔG* hom G L G S G S ΔT r=r* ΔG=2γ/r * r = Volume free energy α r 3 T ΔT T M Note : Both r* and ΔG* depend on undercooling (ΔT). 6

7 Critical Undercooling for Nucleation Assumptions: Liquid with nuclei is an ideal solution of various size clusters. Each size cluster contains i atoms or molecules. Northeastern University Homogeneous nucleation occurs only when liquid is undercooled by T N Critical undercooling for nucleation 7

8 Rate of Homogeneous Nucleation For a given undercooling: Note: C 0, Atoms per unit volume in the liquid. Northeastern University C*, Number of atoms that have reached critical size. Addition of one more atom, converts the clusters to a stable nuclei. If this happens with a frequency of f 0: clusters/m 3 Nuclei / m -3 S -1 N Nuclei / m -3 S -1 ΔT N ΔT No nuclei is formed until ΔT N is reached!! 8

9 Heterogeneous Nucleation In practice, homogeneous nucleation is rarely observed. Sources of nucleation sites: Dislocations Grain boundaries Dust particles Secondary phase particles Mould walls & cracks ΔG het = V(G s G L ) + A SL γ SL + A SM γ SM - A SM γ ML = where, S(θ) 1 is a function of the wetting angle 9

10 ΔG het for a given undercooling (ΔT) ΔG Northeastern University ΔG * hom ΔG * het r * ΔG het r Note: ΔG hom r* depends only on ΔT. ΔG* het depends of S(θ) & ΔT ΔG* het < ΔG* hom 10

11 Variation of ΔG* & nucleation rates with ΔT Smaller undercooling is required for heterogeneous nucleation Nuclei / m -3 S -1 where, f 1 is the frequency factor C 1 is the # of atoms in contact with the heterogeneous nucleation sites. 11

12 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: 12

13 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

14 First order magnetostructural transitions First order magnetostrucutral transitions share common features with solidification. Example: Bulk Fe 1-x Rh x (0.485 < x < 0.55) AFM phase FM phase (Levitin, Soviet Physics JETP, 1966) Phase transition features: Thermal hysteresis T t = f(h,p) (Kouvel and Hartelius, J. Appl. Phys,1962) ~ 1% volume expansion 14

15 Thermal hysteresis Example: Hypothesized FeRh nanoparticles in Cu matrix. Onset of Phase #1 Complete transformation of Phase #2 Complete transformation of Phase #1 Nucleation of Phase #2 T~130 K Type II AFM FM Phase #1: AFM??? PHASE #2: FM??? 15

16 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

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