Precipitates equilibrium shapes

Fig. 10.15. Series of equilibrium shapes as observed in y Cu-Zn (solid hues) and estimated values of the interfacial energy vs orientation (dashed lines) (adapted from Stephens and Purdy (1975)). The precipitate sizes are roughly 60 (im for all but the 500 °C case, in which the precipitates have a size on the order of 500 (im.

Jog C. S., Sankarasubramanian R. and Abinandanan T. A., Symmetry-Breaking Transitions in Equilibrium Shapes of Coherent Precipitates, J. Mech. Phys. Solids, 48, 2363 (2000). 

An alloy is cooled from a temperature at which it has a single-phase structure (a) to a temperature at which the equilibrium structure is two-phase (a -i- ji). During cooling, small precipitates of the P phase nucleate heterogeneously at a grain boundaries. The nuclei are lens-shaped as shown below. 

In a saturated solution, an equilibrium exists between the rate of precipitation of solute particles and the rate of dissolution of solute particles. The rate of precipitation equals the rate of dissolution. The shape of a crystal of solute added to a saturated solution will change after a period of time at a constant temperature and pressure, but its mass will remain the same. The equilibrium between dissolving and precipitation is dynamic, a continuous process. 

Solute movement through soil is a complex process. It depends on convective-dispersive properties as influenced by pore size, shape, continuity, and a number of physicochemical reactions such as sorption-desorption, diffusion, exclusion, stagnant and/or double-layer water, interlayer water, activation energies, kinetics, equilibrium constants, and dissolution-precipitation. Miscible displacement is one of the best approaches for determining the factors in a given soil responsible for the transport behavior of any given solute. 

Metal-Hydroxides. Most heavy metals may precipitate via strong bases (e.g., NaOH and KOH) as metal-hydroxides [M(OH)n]. These precipitation reactions are described in Chapter 2. As noted, metal-hydroxide solubility exhibits U-shape behavior and ideally its lowest solubility point in the pH range allowed by law (e.g., pH 6-9) should be lower than the maximum contaminant level (MCL). However, not all heavy metal-hydroxides meet this condition. The data in Figure 12.1 show the various metal-hydroxide species in solution when in equilibrium with metal-hydroxide solid(s). In the case of Pb2+, its MCL is met in the pH range of 7.4-12, whereas the MCL of cadmium (Cd) the MCL is not met at any pH. Similar information is given by the solubility diagrams of Cu2+, Ni2+, Fe3+ and Al3+. 

Equilibrium for a solid would also stipulate the shape, since the surface free energy should be a minimum. For an isotropic crystal this shape would be that of a sphere, but this is hardly ever a factor. The loss in energy for a surface atom is about one-half of the cohesive energy per atom. But the number of atoms on a typical surface is only about 10, or 10 mol, so the surface energy is very small. There is one observable effect, however a collection of small crystals will cohere to form larger crystals, if a mechanism, such as digestion of a precipitate, is provided. 

Wolverton C., First-Principles Prediction of Equilibrium Precipitate Shapes in Al-Cu Alloys, Phil. Mag. Lett., 79, 683 (1999). 

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