ELEC 474 ME495 ME546 Advanced Topics in Nanotechnology Process Nucleation and Growth Kinetics OutLine Homogeneous nucleation Critical radius nucleation rate Heterogeneous nucleation Nucleation in melting and boiling Growth mechanisms Rate of a phase transformation the earliest stages of crystallization are nucleation processes Nucleation The formation of a new crystalline entity which starts through the nucleation process Nucleation is defined as the series of atomic or molecular processes by which the atoms or molecules of a reactant phase rearrange into a cluster of the product phase large enough as to have the ability to grow irreversibly to a macroscopically larger size Nucleation Nucleation can be Heterogeneous the new phase appears on the walls of the container at impurity particles etc Homogeneous solid nuclei spontaneously appear within the undercooled phase Let s consider solidification of a liquid phase undercooled below the melting temperature as a simple example of a phase transformation Homogeneous Nucleation Is the transition from undercooled liquid to a solid spherical particle in the liquid a spontaneous one That is does the Gibbs free energy decreases Consider a given volume of liquid at a temperature T below Tm with a free energy GI Homogeneous Nucleation When a liquid is cooled below the melting temperature Tm there is a driving force for solidification Gv Gv L GvS 0 Free energy change associated with the formation of a small volume of solid has a negative contribution due to the lower free energy of a bulk solid Interfacial energy SL due to the creation of a solid liquid interface is positive contribution resulting in a kinetic barrier for the nucleation Homogeneous Nucleation The excess free energy associated with the solid particle can be minimized by when solid particle shape is sphere then SL is isotropic and minimized Figure The free energy change associated with homogeneous nucleation of a sphere of radius r Homogeneous Nucleation the interfacial term increases as r2 volume free energy released increases as r3 the creation of small particles of solid always leads to a free energy increase It is this increase that is able to maintain the liquid phase in a metastable state at temperatures below Tm For a given undercooling r is associated with a maximum excess free energy r is the critical nucleus size or critical radius G is the nucleation barrier Figure The free energy change associated with homogeneous nucleation of a sphere of radius r Homogeneous Nucleation If r r the system can lower its free energy by dissolution of the solid Unstable solid particles with r r are known as clusters or embryos when r r the free energy of the system decreases if the solid grows Stable particles with r r are called as nuclei Figure The free energy change associated with homogeneous nucleation of a sphere of radius r Homogeneous nucleation Critical Nucleus Size and Critical Nucleation Barrier Critical nucleus size r and critical nucleation barrier G Homogeneous nucleation critical nucleus size and the critical nucleation barrier The difference between the Gibbs free energy of liquid and solid also called driving force for the phase transformation is proportional to the undercooling below the melting temperature T Tm T Both critical nucleus size r and critical nucleation barrier G decrease with increasing undercooling Homogeneous Nucleation Both critical nucleus size r and critical nucleation barrier G decrease with increasing undercooling At high T small undercoolings r is large and nucleation rate is low At Low T strong undercooling r decrease high nucleation that is nucleation rate is high Rate of Homogeneous Nucleation How fast solid nuclei will appear in the liquid at a given undercooling The rate of nucleation i e the number of nuclei formed per unit time per unit volume depends on formation of a solid nucleus of critical size r and frequency with which atoms from liquid attach to the solid nucleus The nucleation typically is usually controlled by the probability of energy fluctuation sufficient to overcome the activation barrier G r G depends on undercoolingng temperature Rate of Homogeneous Nucleation How fast solid nuclei will appear in the liquid at a given undercooling There is a critical value of undercooling Tcr for nucleation Blow the critical value of undercooling Tcr Until a critical value of undercooling Tcr is achieved nucleation rate increases exponentially Usually Tcr 0 2Tm The large undercoolings are only obtained when no heterogeneous nucleation sites are available i e when solid nuclei must form homogeneously from the liquid Homogenous nucleation can be achieved when very large undercoolings are reached In practice homogeneous nucleation is rarely encountered in solidification Heterogeneous Nucleation The new phase appears on the walls of the container at impurity particles grain boundaries etc G and r depend heavily on the interfacial energies modifies this value will have an effect on the possible viability of the nucleation process so any process that There are three interfacial energies LC liquid container interface LS liquid solid interface SC solid container interface Heterogeneous Nucleation There are three interfacial energies LC liquid container interface LS liquid solid interface SC solid container interface Total interfacial energy of the system is minimized when the embryo has the shape of a spherical cap with a wetting angle Heterogeneous Nucleation The formation of the nucleus leads to excess free energy given by Note that there are now three interfacial energy contributions The first two are positive as they arise from interfaces created during the nucleation process The third however is due to the destruction of the container liquid interface under the spherical cap and results in a negative energy contribution Heterogeneous Nucleation The formation of the nucleus leads to excess free energy given by S is shape factor Except for factor S this expression is the same as that obtained for homogeneous nucleation S has a numerical value 1 dependent only the shape of the nucleus Heterogeneous Nucleation critical nucleus size r and critical nucleation barrier G critical nucleus size r same as for homogeneous nucleation For heterogeneous nucleation the critic nucleus radius r is unaffected by the container wall and only depends on the undercooling Heterogeneous Nucleation critical nucleus size r and critical nucleation barrier G Critical