L5: Precipitate Growth
Learning Objectives:
Understand how the growth rate of a precipitate can be controlled by atoms hopping the interface and consider what interface properties are important
– growth rate is constant with time
– rate varies depending on coherency of interface
Or it can be controlled by diffusion (transport), considering the difference in kinetics between the two
– growth rate is parabolic with time(tends to dominate later in growth)
– depends on undercooling as well (dc/dx).
Discuss how once the precipitates have grown to equilibrium concentration they then coarsen by the large ones growing at the expense of the smaller ones
– smaller than delta Gn (needs higher temp to occur at same rate)
– sets up concentration gradient, so matter diffuses from small to large precipitates
– we never get to equilibrium due to kinetic limitations.
Growth Rate Controlled by Interface Hopping and Interface Properties:
Interfacial Control: When the growth of a precipitate is limited by the rate at which atoms can cross the interface between the matrix and the precipitate, the growth is said to be under interfacial control. The rate of growth is determined by the number of atoms that can hop across the interface per unit area and time.
Key Interface Properties: The mobility of the interface depends on its structure. Incoherent interfaces (disordered) allow faster atom hopping, leading to faster growth, while coherent or semi-coherent interfaces (ordered) slow down growth due to the need for structural reorganization.
Growth Rate: Under interfacial control, the growth rate is constant with time, meaning the radius of the precipitate increases linearly with time.
2. Growth Rate Controlled by Diffusion (Transport):
Diffusional Control: When the growth of a precipitate is limited by the rate at which atoms can diffuse through the matrix to reach the precipitate, the growth is said to be under diffusional control.
Concentration Gradient: The growth rate depends on the concentration gradient of solute atoms (e.g., B atoms) in the matrix. The concentration of solute atoms is highest far from the precipitate and decreases towards the precipitate interface.
Growth Rate: Under diffusional control, the growth rate is parabolic with time, meaning the size of the precipitate increases with the square root of time. This is because atoms must diffuse from increasingly longer distances as the precipitate grows.
Effect of Undercooling: Greater undercooling increases the supersaturation, leading to a steeper concentration gradient and faster growth.
3. Coarsening of Precipitates to Lower Surface Energy:
Coarsening (Ostwald Ripening): Once precipitates have grown to their equilibrium concentration, they undergo coarsening, where larger precipitates grow at the expense of smaller ones. This process reduces the total surface energy of the system.
Driving Force: Smaller precipitates have higher interfacial energy per unit volume compared to larger precipitates. This creates a concentration gradient where solute atoms diffuse from smaller precipitates (higher solubility) to larger ones (lower solubility).
Kinetics: Coarsening is driven by diffusion, and the rate of coarsening depends on temperature. At higher temperatures, diffusion is faster, leading to more rapid coarsening.
Equilibrium: Thermodynamically, coarsening would continue until only one large precipitate remains, but in practice, kinetic limitations (e.g., slow diffusion at low temperatures) prevent the system from reaching this state.
Key Takeaways:
Interfacial Control: Growth rate is constant, determined by interface mobility.
Diffusional Control: Growth rate is parabolic, determined by solute diffusion.
Coarsening: Larger precipitates grow at the expense of smaller ones to reduce surface energy, driven by diffusion and influenced by temperature.