Theory of Binary Systems

The critical concentration at which phase separation first appears on passage to the two-phase region is thus predicted to occur at a very small volume fraction of polymer for a polymer having a molecular weight of a million (a 10 ), for example, y2c=0.01. Substituting Eq. (4) in either of the equations (3), we obtain 

Thus the critical value of xi will exceed 1/2 by a small increment de- 

Before comparing these predictions regarding the critical point with experimental results, we may profitably examine the binodial curve of the two-component phase diagram required by theory. The following useful approximate relationship between the composition V2 of the more dilute phase and the ratio y = V2/v2 of the compositions of the two phases may be derived (see Appendix A) by substituting Eq. (XII-26) on either side of the first of the equilibrium conditions (1), using the notation V2 for the volume fraction in the more dilute phase and V2 for that in the more concentrated phase, and similarly substituting Eq. (XII-32) for fX2 and y,2 in the second of these conditions  

The approximation involved is inconsequential for concentrations up to ten times the critical concentration V2c (see Appendix A). The corresponding value of xi is given by 

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