Nucleation

• Steps in liquid to solid phase transformation:
  1. Liquid
  2. Embryo- tiny solid structures within the liquid form by chance; can form in equilibrium at, above, or below $T_{fr}$
  3. Nuclei- embryos grow into larger pieces as temperature drops
  4. Nucleation- nuclei grow
  5. Growth- nuclei coalesce and form grain boundaries
  6. Solid
• When a liquid is cooled below the freezing temperature, the free energy of the solid is smaller than the liquid, so solids more favorably form than liquids.
  • Change in free energy: $\delta G=\dfrac{4}{3}\pi r^3 \delta G_v+4\pi r^2\sigma_{sl}$
    • Driving energy drop (volumetric): $\delta G_v = \dfrac{Gs-G_l}{V}$; $G_s-G_l<0$ for $T<T_m$
    • Energy cost per surface area to form solid-liquid interface: $\sigma_{sl}$; always >0
    • Sign of $\delta G$ depends on if $\delta G_v$ or $\sigma_{sl}$ is higher
      • for very small r, $\sigma_{sl}$ term wins out (b/c of r^2)
      • for larger r, $\delta G_v$ term wins out (b/c r^3)
  • Because nuclei formation is opposed by the need to create a solid-liquid interface, solidification may not occur at the freezing temperature.
• Nucleation- formation of secondary phase from a primary phase
  • In solidification- the formation of first nanocrystallites from molten metal
• Critical radius ($r^*$)- radius above which particle growth is accompanied by a decrease in free energy
  • Any embryo of $r>r^*$ will form a stable nucleus.
    • How to get to $r^$? —> $\dfrac{d\delta G}{dr} = 0 \rightarrow r^$
      • $r^* = -\dfrac{2\sigma_{sl}}{\delta G_v}$
        • $\delta G_v = - \dfrac{\delta H_f(T_m-T)}{T_m}$
          • $H_f$- heat of fusion, latent heat; amount of heat needed to be absorbed by a material for it to melt
          • the lower T is, the larger the magnitude of $\delta G_v$
      • $r^* = -\dfrac{2\sigma_{sl}T_m}{\delta H_f}\dfrac{1}{(T_m-T)}$
        • Undercooling: $T_m - T$; the more you undercool a material, smaller nuclei are more likely to survive
• $\delta G^*$- energy required to form a stable nucleus (i.e., an infinitely small particle)
  • $\delta G^* = \dfrac{16\pi \sigma_{sl}^3}{3(\delta G_v)^2}$
    • The more you undercool a material, the smaller $\delta G$ will be, so more nuclei will form more quickly
• Homogeneous nucleation- occurs when undercooling is large enough to allow formation of stable nuclei
• Heterogeneous nucleation- relies on impurities in the liquid for nucleation to occur; less (sometimes no) undercooling is needed; big radius of curvature with little solid/liquid contact area
  • Nucleation occurs at contact point of liquid and impurity
  • can achieve large radius without cost of energy to make surface of entire nucleus with larger surface area
  • $\delta G^* = \dfrac{16\pi \sigma_{sl}^3}{3(\delta G_v)^2}f(\theta)$
    • $f(\theta)$ is a constant between 0 and 1; thus, $\delta G^*$ is smaller for heterogeneous nucleation, so more nuclei more easily form

Controlling the Nucleation Process:

• Inoculation- adding small particles to promote nucleation; results in smaller grains and higher strength
  • Each particle acts as a nucleation site. More inoculants means that more grains form. More grains = smaller grains = higher strength.
  • Surface of cast can act as nucleation site.
• Dispersion strengthening- adding fine particles to form a second phase (solid state transformations); results in higher strength
• Metallic glasses- fast cooling prevents nucleation of crystalline structures; results in amorphous structure
• Conclusion: if you can control nucleation, then you can control material properties.

Growth

• Growth depends on how heat is removed from the molten metal.
• First, the specific heat of the liquid must be removed.
  • Step #1: cool liquid to equilibrius freezing/melting temp
• Then, remove latent heat of fusion ($\delta H_f$)
  • How $\delta H_f$ is removed from the solid-liquid interface determines the growth mechanism
• Types of Growth:

1. Planar Growth- well inoculated —> heterogeneous nucleation —> no need for undercooling
   1. A slight bump will form on a surface but will grow at slower rate because it is exposed to liquid at higher temperature; planar interface eventually catches up and levels out with the bump
   2. At liquid-solid interface, $T=T_m$, but temperature in the liquid ahead of the front is greater than $T_m$ —> stable, controlled growth of planar interface as it catches up with bumps
2. Dendritic growth- not well-inoculated —> need undercooling
   1. Temperature in liquid ahead of front is smaller than $T_m$ —> unstable growth of dendrites (tree-like structures) because $T_m-T$ is a driving force of continued bump growth in any direction
      1. primary dendrite —> secondary dendrite
   2. Dendrites don’t like to touch until very late in the solidification process because the surface of dendrites are hot because energy is released as the liquid solidifies, so the liquid between dendrites remains hot until later in the solidification process