SolutionsIf we mix two types of atoms, 3 possibilities for attractive forces,ororIdeal solutions

Vapor pressure of pure metal A = pA*Vapor pressure of A in solution = pAWhere aA is the activity of A metal.Fig. A. Vapor pressure pA of pure metal A in an ideal solution as a function of composition [Alloy Phase equilibria by A. Prince]

In the composition range 0 XA 1, the vapor pressure of A is linear function of composition if the solution is ideal, henceAs aA = pA/pA* for an ideal solution, sopA = XApA* aA = XA This is called Raoults law. The activity of a solvent in an ideal solution is equal to its atomic fraction.

Actual or regular solutionsActual solutions

Nd2Fe14B is the strongest magnet ever produced.

Nucleation: As in solidification, nucleation occurs most easily on surfaces already present in the structure, thereby minimizing the surface energy term. Thus, the precipitates heterogeneously nucleate most easily at grain boundaries and other defects.

Growth: Growth of the precipitates normally occurs by long-range diffusion and redistribution of atoms. Diffusing atoms must be detached from their original locations (perhaps at lattice points in a solid solution), move through the surrounding material to the nucleus, and be incorporated into the crystal structure of the precipitate.

Kinetics The overall rate, or kinetics, of a transformation depends on both nucleationand growth. If more nuclei are present at a particular temperature, growth occurs froma larger number of sites and the phase transformation is completed in a shorter periodof time. At higher temperatures, the diffusion coefficient is higher, growth rates are higher, and again we expect the transformation to be completed in a shorter time, assumingan equal number of nuclei.The rate of transformation is given by the Avrami equation, with the fraction of the transformation, f , related to time, t, by

where c and n are constants for a particular temperature. An incubation time, t0, is the time required for nucleation to occur.

where Q is the activation energy (in this case for the phase transformation), R is the gas constant, T is the temperature, and A is a constant.The rate of atom or ion movement isrelated to temperature or thermal energy by the Arrhenius equation:

Figure 12-3 (a) The effect of temperature on the rate of a phase transformation is the product of the growth rate and nucleation rate contributions, giving a maximum transformation rate at a critical temperature. (b) Consequently, there is a minimum time (tmin) required for the transformation, given by the C-curve.

Widmansttten patternsWidmansttten Structure: The second phase may grow so that certain crystallographic planes and directions in the precipitate are parallel to preferred planes and directions in the matrix, creating a basket-weave pattern known as the Widmansttten structure. This growth mechanism minimizes strain and surface energies and permits faster growth rates. Widmansttten growth produces a characteristic appearance for the precipitate. When a needle-like shape is produced [Figure 12-6(a)], the Widmansttten precipitate may encourage the nucleation of cracks, thus reducing the ductility of the material. However, some of these structures make it more difficult for cracks to propagate, therefore providing good fracture toughness.

Figure 12-6 (a) Widmansttten needles in a Cu-Ti alloy (420). (From ASM Handbook, Vol. 9, Metallography and Microstructure (1985), ASM International, Materials Park, OH 44073.) (b) Continuous y precipitate in an Al-4% Cu alloy, caused by slow cooling (500).