1. Nucleation Don H. Rasmussen Box 5705 Clarkson University rasmu@clarkson.edu

2. Homogeneous Nucleation l Fluctuations in composition and structure which are small in extent but large in degree result in small new phase nuclei which are in local equilibrium but unstable to growth in an undercooled or supersaturated parent phase.

3. How does Homogeneous Nucleation Occur? l Stable clusters form when their formation decreases total free energy. l Growth of small clusters is limited because new particle surface costs more free energy than the bulk free energy reduction. Only large clusters are stable. l Clusters grow and decay by monomer addition/evaporation and there is in a metastable cluster size distribution. l The larger the supersaturation or undercooling, the greater the number and maximum size of the existing clusters and the smaller the necessary critical cluster size for continued growth. l For clusters just larger than critical, the growth rate increases along with the size in an autocatalytic fashion.

4. l Nucleation Rate, J, is the product of n q, the net probability of an atom jumping across interface and into the critical cluster (per unit surface area)  jump frequency of monomer is fluid n A c, the surface area of the critical cluster n n c, the concentration of critical clusters per unit volume Numerical Model of Homogeneous Nucleation

5. q is the net rate of diffusion across the surface of a cluster where D o is the diffusion coefficient in the liquid. and c i and c o are the concentration of crystallizing atoms on the two sides of the interface of thickness, l.

6. The surface area of a spherical cluster is The concentration of critical clusters is

7. The pre-exponential factors are almost constant and approximately 10 35 nuclei/cm 3 sec. Nucleation Rate per Unit Volume, J

8. Effect of Temperature on Bulk Free Energy Change Free energy Temperature  G v L S TT TETE T G L G S

9. Free Energy of a Cluster as a Function of Size

10. Influence of increasing Undercooling or Supersaturation From 1 to 5 the supersaturation or undercooling increases which results in a decrease in both the critical cluster size and the barrier to nucleation.

11. Conditions for critical cluster or nucleus

12. Critical Cluster Size and Free Energy Barrier versus Undercooling

13. Temperature Dependence of Nucleation Rate J(T) TT

14. Time Temperature Transformation Curves  0 200 Time (Seconds) The delay time is related to the reciprocal of the nucleation rate and here the delay time is plotted as a function of the undercooling.

15. Heterogeneous Nucleation l Nucleation at the surface of an impurity particle or on the walls of the container. l “Catalysts for Nucleation” are surfaces which significantly lower the barrier to new phase formation. l Heterogeneous nucleation occurs at low undercooling and at high rates.

16. Nucleation on a substrate takes less material  sl  nl  sn liquid substrate nucleus 

17. Fraction of the critical cluster which must form at any specific undercooling

18. Free Energy of Formation of the Nucleus versus Contact Angle at Fixed Undercooling