Avrami equation

The Johnson - Mehl- Avrami - Kolmogorov equation (short: JMAK equation also Avrami equation) describes the flow of a phase or structural transition at constant temperature ( isothermal change of state). With the aid of the equation is obtained an approximate rate of crystallization. The JMAK equation describes the entire process of transformation with two sizes, the nucleation rate and the rate of growth of already formed areas of the new phase.

Basics

The conversion of one phase to another, for example, the crystallization of an amorphous solid, does not happen everywhere at once, but starts at a few points ( nucleation). From these points of the new phase grows (eg crystallites ). At the same time it comes again and again in other places for nucleation; these areas of the new phase then grow. This happens to all areas of the new phase are finally united and the old phase has completely disappeared. The JMAK equation indicates how large the portion of the new phase of the overall system as a function of time.

Requirement of the behavior described herein is a system consisting of a first phase (hereinafter α ), although a different phase ( β ) is thermodynamically more stable. This occurs, for example, if during cooling of an alloy, the solubility of an element is so low that the alloy is supersaturated, so if more is from this element in the solid state, as it can remain in solution.

The JMAK equation is an important basis for the creation of time - temperature-transformation ( TTT ) diagrams.

Applications

The Johnson - Mehl- Avrami - Kolmogorov equation describes numerous processes in materials science, especially metallurgy, and in physical chemistry:

• Crystallization in an amorphous solid ( eg, polymer).
• Phase transformation with the temperature, for example, when a temperature above a threshold, below a crystal structure is thermodynamically stable.
• In alloys during cooling: formation of precipitates ( precipitates ) of poorly soluble elements or of crystallites with intermetallic phases containing several poorly soluble elements or ( here is, however, described only the beginning of the process because it yes no full conversion of the entire solid comes ).

In many cases, the equation JMAK mainly describes the beginning of the transition well, may occur during the JMAK behavior towards the end of conversion deviations. In the formation of crystals example, this may be related to the fact that different oriented crystals collide and arise between them energetically unfavorable interfaces.

Mathematical treatment

The precipitation of a phase is considered to β from the metastable phase α. Under the assumptions

• Spherical nuclei
• A random distribution of the nuclei in the volume
• A constant nucleation rate N, be formed with the new seeds,
• A constant growth rate v of germs

Yields the fraction f (t ) of the converted structure with time t:

This equation is valid for short and long conversion times t and for small and large conversion shares f:

• For short times, where the particles are still growing independently and where applies, the JMAK equation can be simplified to:

• For long times when it comes to collision of the growing particles or overlap of their diffusion catchment areas, the volume of the converted area increases more slowly than with the fraction f and tends to one:

Both equations are the initial assumptions about embryonic forms and the growth of special cases of a more general law that applies to many other models:

The Avrami exponent n is between 1 and 4; for example, is obtained in two dimensions (crystallization in a very thin layer) an exponent of n = 3

The constant k depends on the nucleation rate and growth rate V N since these depend on the temperature, k is thus also dependent on the temperature: