Description of Phase Transformation

1. Recall the Ehrenfest (“classical”) classification scheme. An nth order transition is defined by the lowest- order derivative which becomes discontinuous at the transition temperature. Description of Phase Transformation (i) Equilibrium phase transformations – occurring at the transition temperature ( e. g. freezing at Tm) (ii) Non-equilibrium phase transformations (freezing below Tm) Equilibrium phase transitions:

2. Modern Scheme of classification: Classifies transitions as either 1st or 2nd order 1st order transition has a latent heat 2nd order transition has no latent heat G G T T Tc Tc For example consider two order – disorder transitions -brass (2nd order) Cu3Au (1st order)

3. H H Latent heat “” transition T T  Cp Cp 1.0 Continuous change in W to Tc W Discontinuous change in W at Tc T T 1.0 W T T Tc Tc Long range order parameter

4. Tid bits • 1st order transition has infinite Cp at Tc • Cp has a lambda point for 2nd order transition • 2nd order (continuous) transitions are studied as the “physics of critical phenomena” Non – equilibrium Transformations: Many important phase transformations do not occur at equilibrium. (e.g. undercooling during solidification) Since these transitions do not occur at Tc, we can’t use the previous classification schemes.

5.    nucleates & grows * motion of a well- defined  /  interface * Criteria for 1st order transition Scheme for classification of non-equilibrium phase transformations: There are two general mechanisms of non-equilibrium transformation. (1) Nucleation and growth - This process is large in degree and small in extent.

6. C r (2) Continuous transformation (e.g., spinodal decomposition) Phase separation occurs by gradual amplification of composition variations. Small in degree and large in extent. * No movement of a sharp interface – 2nd order non-equilibrium phase transformation

7. ’  Homogeneous Nucleation / /  Classical Theory of Nucleation Phase transition requires formation of fluctuation or nucleus.   ’ Barrier - interface critical nucleus size Foreign substances involved which serve as nucleation sites. Heterogeneous Nucleation Heterogeneous Nucleation

8. I (A) Thermodynamics of Nucleation : (i) Homogeneous Nucleation – (1925 Volmer-Weber) Series of bimolecular reactions Here, i* is the critically sized nucleus. Other methods of cluster formation such as simultaneous collisions are less probable. Volmer’s kinetic analysis considered only the forward rate of reaction (A). The back reacting rate was considered small and was ignored

9.  # per unit volume Thermodynamic balance of reactions up to (A) Mixing of ni clusters of size i increases the entropy of the system  some cluster populations are always present. Dilute solution theory for mixture of n1 ……. ni clusters: The chemical potential of an i atom cluster : in equilibrium : (*chemical potential per mole the same in all clusters  likely an incorrect assumption!)

10. (i.e., # of clusters of i 1 is small) Approximation : where

11. volume of α’ area assumed independent of i and positive negative The nucleation of ’ creates an interface which costs energy. The free energy of nucleus formation, Gi, is generally composed of 2 terms : (a) A volume or bulk term describing the free energy change driving the transformation (b) an interfacial energy term opposing the transformation. These assumptions fit a central force nn bond model each atom has Z bonds of energy   bulk energy of solid = -1/2 Z per atom

12. 4r2 G G* r r* Gv 4/3r3 for spherical clusters : critical nucleus :

13. The equilibrium conc. of critical nuclei What to use for Gv? Pure materials Gas pressure • Gas  Liquid Gas  Solid Equil. vap pressure (2) Condensed phases Entropy of transformation Recall: liq  sol ; Te = equil. transition temp.

14. T* Tm Thermodynamic Description of Phase Transitions 1. Component Solidification G @ T= Tm Gsolid Gl =Gs dGl = dGs ; G = 0 Gliquid T Where L is the enthalpy change of the transition or the heat of fusion (latent heat).

15. Alloys:a’nucleus can have a composition different froma. e.g., binary system, x is atom fraction of one element.

16. (ii) Heterogeneous Nucleation –  -phase   aS S S-substrate  contact angle s = bS - aS

17. Superhydrophobic surface – Lotus leaf

18.  = 0 Perfect wetting  90  90 = 180 No wetting For  >  : No wetting  foreign substrate does not help nucleation

19. If surface deforms    S  θ ,  S Determination of free energy of formation (GS) of a nucleus on a substrate: Shape : assume spherical cap of radius r, contact angle  Volume of cap:   r

20. Contact circle area Cap area bS - aS Cap volume

21. # of single atom sites on substrate (no undercooling) (no help from foreign substrate) Concentration of nuclei