Question

a) DERIVE the equation for the change in the free energy of a CYLINDRICAL nucleus heterogeneously forming from an existing phase. The nucleus grows with its circular base on

the mold wall and the axis of the cylinder normal to the wall. ASSUME that the height of the cylinder can be written as h = a * r, where a is a constant, and r is the cylinder radius.

b) Find the critical radius and the energy barrier to nucleation for this shape of precipitate.

c) We can vary the shape of the nucleating phase by changing the constant a. Make a log- log plot of the size of the energy barrier versus the parameter a over a range from a

= 0.01 (disk shaped) to a = 100 (needle shaped). Explain your results.

Answer 1

consider the cylindrical embryo with radius r and height h, let      be the interfacial energies b/w vapour liquid, the liquid substrate(wall) and vapour substate and the interfacial energy of the curved surface so the free energy of formation is,

  

given h=a r,

taking the rate of change with respect to r

  

   

critical radius r_c is when the above expression is equal to zero

  

  

putting r_c in equation we get critical energy

if we plot a graph b/w the above two parameters we get a linear plot passing through the origin.so as a increases the free energy barrier also increases linearly, so creating a needle-shaped nucleus will require more energy so it's not favourable, ad disc-shaped one is favoured by the system.