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Eutectic Solidification

  Extremum Growth     7 of 9
It has become clear that, for a given undercooling DT0, various combinations of interlamellar spacing l and growth velocity V are possible. However, there is a particular spacing which gives a maximum velocity. This can be found by evaluating dV/dl and setting it equal to zero.

extremum_eq1.gif (585 bytes)

When a spacing is adopted which gives this maximum velocity, growth is said to be occurring under extremum conditions. It can be seen that the extremun spacing is given by le = 2lmin  
This is also the spacing of which, for a fixed velocity, growth would be taking place with the minimum interfacial undercooling.
In practice, most eutectic systems tend to solidify at the extremum. Substitution of the above expression for l into the equation relating V to l now gives

extremum_eq2.gif (282 bytes)

which, using the relationship between lmin and DT0 can be written as

extremum_eq3.gif (356 bytes)

or, expressed in terms of the expected spacing l (=le)

extremum_eq4.gif (351 bytes)

The RHS of this equation is a constant for a given system. It typically has a value of about 10-16 m3 s-1. Clearly faster growth leads to finer lamellar spacings.

  Kinetics | Redistribution | Cell, dendrite and grain structure | Eutectic

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