GTBL042-10 GTBL042-Callister-v2 August 13, 2007 18:16
10.9 Development of Microstructure in Isomorphous Alloys • 351For multiphase alloys, it is often more convenient to specify relative phase
amount in terms of volume fraction rather than mass fraction. Phase volume frac-
tions are preferred because they (rather than mass fractions) may be determined
from examination of the microstructure; furthermore, the properties of a multiphase
alloy may be estimated on the basis of volume fractions.
For an alloy consisting ofαandβphases, the volume fraction of theαphase,Vα,
is defined asVα=vα
vα+vβ(10.5)
αphase volume
fraction—
dependence on
volumes ofαandβ
phases
wherevαandvβdenote the volumes of the respective phases in the alloy. Of course,
an analogous expression exists forVβ, and, for an alloy consisting of just two phases,
it is the case thatVα+Vβ=1.
On occasion conversion from mass fraction to volume fraction (or vice versa) is
desired. Equations that facilitate these conversions are as follows:Vα=Wα
ρα
Wα
ρα+
Wβ
ρβ(10.6a)Conversion of mass
fractions ofαandβ
phases to volume
fractionsVβ=Wβ
ρβ
Wα
ρα+
Wβ
ρβ(10.6b)andWα=Vαρα
Vαρα+Vβρβ(10.7a)
Conversion of
volume fractions ofα
andβphases to mass
fractions Wβ= Vβρβ
Vαρα+Vβρβ(10.7b)In these expressions,ραandρβare the densities of the respective phases; these may
be determined approximately using Equations 5.13a and 5.13b.
When the densities of the phases in a two-phase alloy differ significantly, there
will be quite a disparity between mass and volume fractions; conversely, if the phase
densities are the same, mass and volume fractions are identical.10.9 DEVELOPMENT OF MICROSTRUCTURE
IN ISOMORPHOUS ALLOYS
Equilibrium Cooling
At this point it is instructive to examine the development of microstructure that oc-
curs for isomorphous alloys during solidification. We first treat the situation in which
the cooling occurs very slowly, in that phase equilibrium is continuously maintained.