Volume Fraction of Phases

Information on the relative quantities of the various phases in the microstructure is also often desirable. These data are most frequently expressed as the volume fraction, Vy, of the features of interest within the microstructure. If one assumes a random sample without substantial anisotropy, one conveniently finds that the volume fraction is equal to the hneal fraction described previously, as well as to the area fraction Ayi and point fraction P/ Vy = = Lj = Pp- 

One of the primary goals of modern materials science is to develop desirable microstructures that provide superior performance through controlled processing. This chapter provides an overview of many features that contribute to the overall microstructure of ceramic materials. In addition to the three-dimensional details of phase distribution that are commonly considered, other structural components including interfaces and defects may play important roles in determining the materials properties. The morphology on a microscopic scale is therefore of keen interest in the study of ceramic properties. 

for example, d) R. K. Watts. Point Defects in Crystals. Wiley, New York, 1976, and b) R. J. D. Tilley. Defect Crystal Chemistry and Its Applications. Chapman and Hall, New York, 1987. 

5 For example, see papers in Science and Technology ofZirconia II. Advances in Ceramics, Vol. 12. (N. Claussen, M. Ruhle, and A. H. Heuer, Eds.) The American Ceramic Society, Columbus, OH, 1984. 

8 Epitaxial Growth. (J. W. Matthews, Ed.) Academic Press, New York, 1975. 

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Vs) is the total volume of stationary phase in the column and (a) is the volume fraction of phase (A) in the stationary phase mixture... 

Rg = volume fraction of phase, dimensionless, gas phase Rl = 1 R g volume fraction of phase, dimensionless, liquid phase... 

L tj) or, factor from Table 10-46 or, mean radius of bend, in. or, rcQux raBo, mol condensate returned/mol product withdrawn or, volume fraction of phase, dimensionless. 

Different components of the mixed phase occupy different volumes of space. To describe this quantitatively, we introduce the volume fraction of normal quark matter as follows X sc = Vnq/V (notation Xb means volume fraction of phase A in a mixture with phase B). Then, the volume fraction of the 2SC phase is given by X2nq = (1 — X.2sc)- From the definition, it is clear that 0[Pg.236]

Quantities useful for predicting phase continuity and inversion in a stirred, sheared, or mechanically blended two-phased system include the viscosities of phases 1 and 2, and and the volume fractions of phases 1 and 2, and ij. (Note These are phase characteristics, not necessarily polymer characteristics.) A theory was developed predicated on the assumption that the phase with the lower viscosity or higher volume fraction will tend to be the continuous phase and vice versa (23,27). An idealized line or region of dual phase continuity must be crossed if phase inversion occurs. Omitted from this theory are interfacial tension and shear rate. Actually, low shear rates are implicitly assumed. 

The composition of an alloy can be found from the volume fractions of phases. The relative weight of component B in the ot phase is (Va)(pa)(Ca), where Va is... 

Hence, the phase average can be related to the intrinsic average by the volume fraction of phase k, c, as... 

V, and V2 are volume fractions of phases 1 and 2, representing carbon nanotubes and carbon fibers respectively. 

The volume fractions of phases in a structure can be very important (Figure Cl-11). In a system with two phases the two volume fractions add up to one. The density of the stmcture is equal to the sum of the products of (volume fraction times density) of the pure phases. If you know these pure densities, you can determine the volume fractions from a measurement of the density of the structure. 

When one adds phase B quicker, or agitates the system less, the droplets of B will even have inclusions of phase A. This is shown in Figure 15.23(b). Such an emulsion is much less stable against inversion, which will take place at lower volume fractions of phase B. In these circumstances, the system shows much less hysteresis. 

Ideal gas universal constant Volume fraction of phase i Reynolds number Density ratio... 

The concept of volume fraction is introduced here heuristically without resorting to a rigorous treatment. With this approach, it is assumed that it is meaningful to conceive a volume fraction of phase k, in any small volume of space at any particular time. If there are n phases in total, this gives ... 

Suppose we wish to determine the amount of phase A in a mixture of phases A, B, C,..., where the relative amounts of the other phases present (B, C, D,...) may vary from sample to sample. With a known amount of original sample we mix a known amount of a standard substance S to form a new composite sample. Let Ca and be the volume fractions of phase A in the original and composite samples, respectively, and let be the volume fraction of S in the composite sample., If a diffraction pattern is now prepared from the composite sample, then from Eq. (14-2) the intensity of a particular line from phase A is given by... 

Fig. 13. Coexistence, hysteresis and kinetics, (a) Schematic of interface-controlled growth of phase ft into phase a (b) superposition of spectra and thermal hysteresis (c) time evolution of volume fraction of phase ft Avrami model (Eq. (14)) for different exponents n.

The volume fraction of phase 1 excluding the transition zone is now equal to 0i — e/2, and the volume fraction of phase 2 is equal to 02 — e/2. We therefore have... 

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

Figure 6. (o) Volume fraction of phases as a function of temperature in Ndo.sSro.sMnOs- Open diamonds, open circles and open squares represent ferromagnet (FM), antiferromagnet (A-type, A-AF) and antiferromagnet (CE-type, CO-CEAF), respectively, (ft) Phase fractions at 125 K at different magnetic fields (i) 0 T and (ii) 6 T (iii) and (iv) show the magnetic moment in each of the phases ((o) after Woodward et al. (1999) and (ft) after Ritter et al. (2000)). 

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